All practice questions

Question 101Question →

The following sentences are from a paragraph about the preservation of ancient scrolls. Arrange the sentences in the most logical order to form a coherent paragraph that begins with a clear introduction and ends with an effective concluding sentence.

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Answer

The sentences should be ordered as follows: first, the sentence introducing the carbonized scrolls of Herculaneum; second, the sentence describing early physical attempts to unroll them; third, the sentence introducing multi-spectral imaging; fourth, the sentence explaining how infrared light reveals the script; and finally, the sentence summarizing the preservation and restoration of the texts.

The correct ordering begins with the sentence introducing the carbonized scrolls at Herculaneum, which sets the historical context. It is followed by the sentence describing the centuries of destructive physical attempts to unroll them. The third sentence introduces multi-spectral imaging to overcome 'this barrier' (the destruction). The fourth sentence explains the mechanics of 'this technology' using infrared light. The final sentence, beginning with 'Ultimately,' provides a strong conclusion that summarizes the dual benefits of preservation and restoration.

Step-by-Step Solution

1

Identify the opening sentence that introduces the main topic.

The sentence about the Mount Vesuvius eruption and the carbonized, unreadable scrolls establishes the context.

A coherent paragraph must begin with an introduction of the subject before describing subsequent actions or technologies.

2

Determine the logical sequence of the middle sentences using transitional cues.

The sentence about 'early attempts to unroll these scrolls' follows the introduction, and the sentence beginning 'To overcome this barrier' must follow the destructive attempts. The sentence about 'this technology' must follow the introduction of multi-spectral imaging.

Pronouns and demonstrative adjectives like 'this barrier' and 'this technology' must immediately follow the specific nouns they refer to.

3

Identify the concluding sentence that wraps up the paragraph.

The sentence starting with 'Ultimately' summarizes the overall impact (preservation and restoration of literature).

An effective conclusion synthesizes the paragraph's main points and highlights the broader significance of the topic.

Key Concept

Opening and Concluding Sentences

Estimated Time:2m 0s

Question 102Question →

The writer wants to organize the sentences below to logically present the chronological steps and historical significance of the accidental discovery of YInMn Blue. What is the most logical order for these sentences?

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Answer

The correct sequence begins with the introduction of the 2009 experiment, followed by the description of the heat reaction, then the realization of the pigment's durability, and concludes with its naming and historical significance.

The correct order follows a clear chronological and logical progression. It starts by introducing the initial experiment in 2009, followed by the immediate reaction to the heating process, then details the physical properties observed by the chemist, and concludes with the naming and historical impact of the pigment.

Step-by-Step Solution

1

Identify the introductory sentence.

The sentence introducing the initial 2009 experiment by Mas Subramanian must come first.

It establishes the timeline, the research group, and the original objective of the study.

2

Identify the direct outcome of the experiment.

The sentence describing the surprise transformation under intense heat must follow.

It details what actually happened during the experiment described in the first step.

3

Determine the immediate evaluation of the outcome.

The sentence about Subramanian's realization of the unique crystal structure is placed third.

It explains the next logical step: testing and analyzing the properties of the unexpected blue material.

4

Identify the concluding summary.

The sentence naming the pigment 'YInMn Blue' and describing its historical context goes last.

It wraps up the narrative by naming the creation and summarizing its long-term significance.

Key Concept

Organizing sentences to establish chronological coherence and meet rhetorical goals.

Question 103Question →

A student is writing a paragraph about the invention of the chocolate chip cookie. Arrange the following sentences in the most logical order to create a coherent paragraph that begins with a clear introduction and ends with an effective conclusion.

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Answer

The correct order begins with the introduction of Ruth Wakefield's accidental creation, followed by her running out of baker's chocolate, then her substituting chopped semi-sweet chocolate, and concluding with the chocolate pieces remaining intact to create a new favorite treat.

The correct order follows a logical and chronological flow. The paragraph begins with an introduction to Ruth Wakefield and her creation. The middle sentences describe the chronological process of running out of baker's chocolate and substituting semi-sweet chocolate. The final sentence wraps up the narrative by detailing the final outcome of the baking process, serving as a clear conclusion.

Step-by-Step Solution

1

Identify the introductory sentence.

The sentence introducing Ruth Wakefield and her famous accidental creation is identified as the opening sentence.

An effective paragraph must begin with an introduction that establishes the main subject before describing specific events.

2

Establish the chronological order of the body sentences.

The sentence about running out of baker's chocolate is followed by the sentence explaining the substitution of semi-sweet chocolate.

The narrative must flow logically from identifying a problem to taking an action to resolve it.

3

Identify the concluding sentence.

The sentence describing the final baked result and its lasting status as a staple is placed at the end.

An effective conclusion resolves the narrative conflict and summarizes the overall outcome or significance of the topic.

Key Concept

Opening and Concluding Sentences

Question 104Question →

The following paragraphs are from an essay about Marie Tharp's mapping of the ocean floor. To make the essay flow most logically, in what order should the paragraphs be placed?

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Answer

The paragraphs should be placed in the order: first, the paragraph describing the historical assumption of a flat ocean floor; second, the paragraph introducing Marie Tharp's sonar analysis and creation of topographic profiles; and third, the paragraph explaining the impact of this groundbreaking map on continental drift theory.

The correct sequence begins with the paragraph establishing the historical belief that the seafloor was a featureless plain. The next paragraph transitionally shifts focus using 'however' to describe Marie Tharp's mid-1950s research and her creation of topographic profiles. The final paragraph logically follows by using 'This groundbreaking map' to refer back to Tharp's profiles and explain the map's role in supporting plate tectonics.

Step-by-Step Solution

1

Identify the introductory paragraph that establishes the context or baseline belief.

The paragraph starting with 'For centuries' describes the historical view that the ocean floor was a flat, featureless plain.

This establishes the status quo before any discoveries were made.

2

Look for transition words or phrases that signal a shift away from this status quo.

The paragraph starting with 'In the mid-1950s, however' uses the contrast transition 'however' to introduce Marie Tharp's research and the discovery of a mountain range, contradicting the flat seafloor theory.

The word 'however' directly connects to and refutes the idea of a featureless seabed established in the first paragraph.

3

Identify the paragraph that builds upon the details of the discovery.

The paragraph starting with 'This groundbreaking map' references the map created in the previous step ('topographic profiles') and explains its larger scientific implications.

The demonstrative adjective 'This' in 'This groundbreaking map' requires a prior mention of mapmaking, which occurs at the end of the second paragraph.

Key Concept

Paragraph-Level Transitions

Question 105Question →

The following sentences are from a paragraph about Polynesian navigation techniques. Arrange the sentences in the most logical order to create a cohesive paragraph that begins with the most effective introduction and ends with the most effective concluding sentence.

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Answer

The correct order of the sentences is: the introduction introducing Polynesian navigators, followed by the creation of stick charts, then the explanation of how they were memorized on land, and finally the concluding sentence summarizing their overall effectiveness.

The correct order establishes a logical progression: first introducing the general topic of instrument-free navigation, then defining the alternative tool used ('stick charts'), next explaining the procedural constraint of memorizing these charts, and finally concluding with a statement on their overall utility and success.

Step-by-Step Solution

1

Identify the most general sentence that introduces the main topic without requiring prior context.

The sentence about Polynesian navigators traversing the Pacific without Western instruments is identified as the opening sentence.

An effective introduction must establish the subject matter and tone of the paragraph without relying on preceding sentences for clarity.

2

Look for transition words or referential pronouns that link to the opening sentence.

The sentence mentioning 'these featureless seas' and introducing 'stick charts' is placed second.

The phrase 'these featureless seas' directly references the 'open waters of the Pacific Ocean' from the first sentence, creating a logical transition.

3

Find the sentence that elaborates on the newly introduced concept of 'stick charts'.

The sentence about how the charts were fragile and memorized on land is placed third.

The term 'these fragile wooden diagrams' connects back to the physical description of 'stick charts' in the second sentence.

4

Determine which sentence provides a summarizing or concluding thought for the paragraph.

The sentence starting with 'By translating the dynamic currents...' is placed last.

This sentence sums up the function and success of the navigational aids, resolving the paragraph's main theme.

Key Concept

Opening and Concluding Sentences

Estimated Time:1m 30s

Question 106Question →

The following paragraphs are from an essay about the history of the pencil. To make the essay flow most logically, in what order should the paragraphs be placed?

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Answer

The paragraphs should be ordered: first the paragraph describing the discovery of graphite in 1564, followed by the paragraph addressing the practical challenges of raw graphite and the introduction of wooden casings, and finally the paragraph discussing the limitations of early casings and the 1795 clay-mixture solution.

The correct order follows a logical chronological and thematic progression. The paragraph detailing the 1564 discovery of graphite establishes the context. The paragraph beginning with 'Despite its usefulness, this raw graphite...' transitions by explaining the drawbacks of the raw mineral and introduces wooden casings. The paragraph beginning with 'These early wooden casings...' uses a clear transitional phrase to link back to the casings and explains the final refinement in 1795.

Step-by-Step Solution

1

Identify the introductory paragraph by searching for independent context that does not rely on prior information.

The paragraph describing the 1564 discovery of solid graphite in Borrowdale, England contains no transitional pronouns or references, establishing it as the starting point.

An essay must introduce the subject and historical origin before detailing later refinements or problems.

2

Link the next paragraph by matching the physical drawbacks of the newly discovered material to its initial use.

The paragraph starting with 'Despite its usefulness, this raw graphite...' connects directly to the raw mineral's use in marking sheep and introduces the wooden casing solution.

The transition 'Despite its usefulness' bridges the discovery of the material with its practical handling challenges.

3

Place the concluding paragraph by identifying the transition that references the casing solution.

The paragraph starting with 'These early wooden casings...' follows the introduction of the wooden sleeves and concludes with the 1795 kiln-firing innovation.

The demonstrative phrase 'These early wooden casings' requires the previous paragraph to have already introduced the wooden sleeves.

Key Concept

Paragraph-Level Transitions

Estimated Time:1m 30s

Question 107Question →

Evaluate each of the following mathematical expressions using the standard order of operations. Arrange the expressions in order from least to greatest value.

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Answer

The correct order of the expressions from least to greatest value is:

2 - 3 \times 4 - (6 + 5)

(evaluates to

-21

),

(2 - 3) \times 4 - (6 + 5)

(evaluates to

-15

),

2 - (3 \times 4 - 6) + 5

(evaluates to

1

), and

2 - 3 \times (4 - 6) + 5

(evaluates to

13

).

Evaluating the four expressions using the correct order of operations (parentheses first, then multiplication, and finally addition/subtraction from left to right) gives the values

-21

,

-15

,

1

, and

13

. Arranged from least to greatest, the correct order is

2 - 3 \times 4 - (6 + 5)

followed by

(2 - 3) \times 4 - (6 + 5)

, then

2 - (3 \times 4 - 6) + 5

, and finally

2 - 3 \times (4 - 6) + 5

.

Step-by-Step Solution

1

Evaluate the expression

2 - 3 \times 4 - (6 + 5)

.

-21

First evaluate the operations in parentheses:

6 + 5 = 11

. Then perform multiplication:

3 \times 4 = 12

. The expression is now

2 - 12 - 11

. Perform subtraction from left to right:

2 - 12 = -10

, and

-10 - 11 = -21

.

2

Evaluate the expression

(2 - 3) \times 4 - (6 + 5)

.

-15

First evaluate the operations in parentheses:

2 - 3 = -1

and

6 + 5 = 11

. The expression is now

-1 \times 4 - 11

. Perform multiplication:

-1 \times 4 = -4

. Finally, perform subtraction:

-4 - 11 = -15

.

3

Evaluate the expression

2 - (3 \times 4 - 6) + 5

.

1

First evaluate inside the parentheses, performing multiplication before subtraction:

3 \times 4 = 12

, and

12 - 6 = 6

. The expression is now

2 - 6 + 5

. Perform subtraction and addition from left to right:

2 - 6 = -4

, and

-4 + 5 = 1

.

4

Evaluate the expression

2 - 3 \times (4 - 6) + 5

.

13

First evaluate the subtraction in parentheses:

4 - 6 = -2

. The expression is now

2 - 3 \times (-2) + 5

. Next, perform multiplication:

-3 \times (-2) = 6

. Finally, perform addition from left to right:

2 + 6 + 5 = 13

.

5

Order the evaluated results from least to greatest.

-21 < -15 < 1 < 13

Comparing the values:

-21

is the smallest value, followed by

-15

, then

1

, and

13

is the largest value.

Key Concept

Evaluating arithmetic expressions containing multiple operations using PEMDAS order of operations.

Question 108Question →

Evaluate the following three mathematical expressions using the standard order of operations, and arrange them in order from least to greatest value.

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Answer

The correct order of the expressions from least to greatest value is

(12 - 4) \div 2

(value of

4

), followed by

12 \div (4 - 2)

(value of

6

), and finally

12 - 4 \div 2

(value of

10

).

Evaluating the expressions according to the order of operations results in values of

10

,

4

, and

6

. Since

4 < 6 < 10

, the correct order from least to greatest value is

(12 - 4) \div 2

, then

12 \div (4 - 2)

, and lastly

12 - 4 \div 2

.

Step-by-Step Solution

1

Evaluate

12 - 4 \div 2

.

The value is

10

.

Division has higher precedence than subtraction, so divide

4

by

2

first, then subtract from

12

.

2

Evaluate

(12 - 4) \div 2

.

The value is

4

.

Parentheses have the highest precedence, so evaluate the subtraction inside first, then perform the division.

3

Evaluate

12 \div (4 - 2)

.

The value is

6

.

Evaluate the subtraction inside the parentheses first, then divide

12

by the result.

4

Compare the calculated values to order the expressions from least to greatest.

The order of values is

4 < 6 < 10

.

Arranging the evaluated values in ascending order gives the correct sequence of the expressions.

Key Concept

The standard order of operations (PEMDAS) requires operations in parentheses to be performed first, followed by multiplication and division from left to right, and then addition and subtraction from left to right.

Question 109Question →

Evaluate the following three mathematical expressions and arrange them in order from least to greatest value.

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Answer

The correct order of the expressions from least to greatest value is

4 \times (2 - 3)

, followed by

4 - 2 \times 3

, and then

(4 - 2) \times 3

.

Evaluating each expression using the order of operations results in values of

-2

,

6

, and

-4

. Arranging these values in increasing order yields

-4 < -2 < 6

, which corresponds to the sequence

4 \times (2 - 3)

, followed by

4 - 2 \times 3

, and then

(4 - 2) \times 3

.

Step-by-Step Solution

1

Evaluate the expression

4 - 2 \times 3

.

-2

Apply the order of operations (PEMDAS) by performing the multiplication first:

2 \times 3 = 6

. Then perform the subtraction:

4 - 6 = -2

.

2

Evaluate the expression

(4 - 2) \times 3

.

6

Apply the order of operations by simplifying the subtraction inside the parentheses first:

4 - 2 = 2

. Then perform the multiplication:

2 \times 3 = 6

.

3

Evaluate the expression

4 \times (2 - 3)

.

-4

Apply the order of operations by simplifying the subtraction inside the parentheses first:

2 - 3 = -1

. Then perform the multiplication:

4 \times (-1) = -4

.

4

Compare the evaluated values to order them from least to greatest.

-4 < -2 < 6

Comparing the values reveals that

-4

is the smallest value,

-2

is the middle value, and

6

is the largest value.

Key Concept

Order of operations (PEMDAS) requires performing calculations in parentheses first, followed by exponents, then multiplication and division (from left to right), and finally addition and subtraction (from left to right).

Question 110Question →

Four mathematical expressions are shown below:

* Expression 1: $\frac{4 \times (5 - 3)^3}{2^2} - 15 \div 3$
* Expression 2: $|-8 - 2| \times (3 - 5)^2 \div 8$
* Expression 3: $24 - 4 \times 3 + 2^3 \div 4$
* Expression 4: $3 \times (4 - 7)^2 - | -2^4 + 6 |$

Arrange these expressions in order from least to greatest value.

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Answer

Expression 1, Expression 2, Expression 3, Expression 4

Evaluating each expression using the correct order of operations yields the values 3, 5, 14, and 17, respectively. Ordering these values from least to greatest gives the sequence of Expression 1, Expression 2, Expression 3, and Expression 4.

Step-by-Step Solution

1

Evaluate Expression 1:

\frac{4 \times (5 - 3)^3}{2^2} - 15 \div 3

3

First simplify the expression inside the parentheses:

5 - 3 = 2

. Next, evaluate the exponents:

2^3 = 8

and

2^2 = 4

. Then perform multiplication and division from left to right:

\frac{4 \times 8}{4} = \frac{32}{4} = 8

and

15 \div 3 = 5

. Finally, perform subtraction:

8 - 5 = 3

.

2

Evaluate Expression 2:

|-8 - 2| \times (3 - 5)^2 \div 8

5

Simplify inside the absolute value brackets and parentheses first:

-8 - 2 = -10

, and its absolute value is

|-10| = 10

. Also,

3 - 5 = -2

. Next, evaluate the exponent:

(-2)^2 = 4

. Then perform multiplication and division from left to right:

10 \times 4 = 40

, and

40 \div 8 = 5

.

3

Evaluate Expression 3:

24 - 4 \times 3 + 2^3 \div 4

14

Evaluate the exponent first:

2^3 = 8

. Next, perform multiplication and division from left to right:

4 \times 3 = 12

and

8 \div 4 = 2

. Finally, perform addition and subtraction from left to right:

24 - 12 + 2 = 12 + 2 = 14

.

4

Evaluate Expression 4:

3 \times (4 - 7)^2 - | -2^4 + 6 |

17

Simplify inside the parentheses:

4 - 7 = -3

, and evaluate its square:

(-3)^2 = 9

. Then multiply:

3 \times 9 = 27

. For the absolute value expression, evaluate the exponent first:

-2^4 = -(2^4) = -16

. Add:

-16 + 6 = -10

, and take its absolute value:

|-10| = 10

. Finally, perform subtraction:

27 - 10 = 17

.

5

Compare the evaluated values to order the expressions from least to greatest.

3 < 5 < 14 < 17

Comparing the values gives the sequence: Expression 1 (value of 3), Expression 2 (value of 5), Expression 3 (value of 14), and Expression 4 (value of 17).

Key Concept

Order of Operations (PEMDAS)

Question 111Question →

Evaluate the four mathematical expressions below and arrange them in order from least to greatest value.

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Answer

Expression X, Expression W, Expression Z, Expression Y

Evaluating all four expressions using the proper order of operations (PEMDAS) yields the values: Expression X is

-16

, Expression W is

-10

, Expression Z is

6

, and Expression Y is

7

. Ordering these values from least to greatest results in the sequence Expression X, Expression W, Expression Z, Expression Y.

Step-by-Step Solution

1

Evaluate Expression X:

-(-3)^2 - | -6 \times 2 + 5 |

-16

First, evaluate the exponent:

(-3)^2 = 9

, which makes the first term

-9

. Next, calculate the expression inside the absolute value bars:

-6 \times 2 + 5 = -12 + 5 = -7

. The absolute value of

-7

is

7

. Subtracting this value gives

-9 - 7 = -16

.

2

Evaluate Expression W:

-2^4 - 3(5 - 7)^2 \div (-2)

-10

First, evaluate the exponent:

-2^4 = -(2^4) = -16

. Then, evaluate inside the parentheses:

5 - 7 = -2

, and square it:

(-2)^2 = 4

. Next, perform multiplication and division from left to right:

-3(4) \div (-2) = -12 \div (-2) = 6

. Adding this to

-16

gives

-16 + 6 = -10

.

3

Evaluate Expression Z:

\frac{-18 \div (-3) \times (-2)}{-4 + 2^3 \div 4}

6

In the numerator, perform multiplication and division from left to right:

-18 \div (-3) = 6

, and then

6 \times (-2) = -12

. In the denominator, evaluate the exponent:

2^3 = 8

, divide:

8 \div 4 = 2

, and add:

-4 + 2 = -2

. Finally, divide the numerator by the denominator:

\frac{-12}{-2} = 6

.

4

Evaluate Expression Y:

4 - 3 \times [8 - (2 - 5)^2]

7

Evaluate the innermost parentheses first:

2 - 5 = -3

, and square it:

(-3)^2 = 9

. Simplify inside the brackets:

8 - 9 = -1

. Multiply:

-3 \times (-1) = 3

. Add to the remaining term:

4 + 3 = 7

.

5

Compare the evaluated values to find the correct ordering.

Expression X (

-16

) < Expression W (

-10

) < Expression Z (

6

) < Expression Y (

7

)

Comparing the values

-16

,

-10

,

6

, and

7

from least to greatest establishes the correct sequence.

Key Concept

Order of operations dictates evaluating parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right). Negation outside exponents and absolute values must be treated with care.

Question 112Question →

Let the values of three mathematical expressions be represented by $A$ , $B$ , and $C$ , where:

A = -3^2 - ( -2 - 4 )^2 \div 3 \times 2

B = |-5 - (-3)| - 2^4 \div (-2)^3 \times (4 - 7)

C = \frac{4 - 3 \times 2^2}{2} - ( -1 )^5 \times 3

Arrange the expressions $A$ , $B$ , and $C$ in order from least to greatest numerical value.

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Answer

Expression A, Expression B, Expression C

Evaluating each expression using correct PEMDAS order of operations yields

A = -33

,

B = -4

, and

C = -1

. Comparing these values gives the order from least to greatest as Expression A, followed by Expression B, followed by Expression C.

Step-by-Step Solution

1

Evaluate Expression A by applying order of operations.

A = -33

Simplify the grouping

(-2 - 4) = -6

. Then apply exponents:

-3^2 = -9

and

(-6)^2 = 36

. Perform division and multiplication from left to right:

36 \div 3 \times 2 = 12 \times 2 = 24

. Subtract to find

-9 - 24 = -33

.

2

Evaluate Expression B by applying order of operations.

B = -4

Simplify grouping terms:

|-5 - (-3)| = 2

and

(4 - 7) = -3

. Evaluate exponents:

2^4 = 16

and

(-2)^3 = -8

. Perform division and multiplication from left to right:

16 \div (-8) \times (-3) = -2 \times (-3) = 6

. Subtract:

2 - 6 = -4

.

3

Evaluate Expression C by applying order of operations.

C = -1

Simplify the numerator:

4 - 3 \times 2^2 = 4 - 12 = -8

, so the fraction is

-4

. Evaluate the exponent:

(-1)^5 = -1

. Multiply:

-1 \times 3 = -3

. Subtract:

-4 - (-3) = -1

.

4

Compare the resulting values to determine the correct order from least to greatest.

Expression A (

-33

) < Expression B (

-4

) < Expression C (

-1

)

Comparing negative integers,

-33

is the smallest value and

-1

is the largest value.

Key Concept

Order of Operations and Number Properties

Question 113Question →

Given the algebraic expression $A = x - y \times z^2$ , evaluate the expression for each of the following sets of values for $(x, y, z)$ and arrange the sets in order of their resulting values of $A$ from least to greatest.

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Answer

The correct order of the sets from least to greatest is:

(x, y, z) = (-5, 1, 2)

resulting in

-9

, followed by

(x, y, z) = (-2, 3, -1)

resulting in

-5

, then

(x, y, z) = (8, 4, 0)

resulting in

8

, and finally

(x, y, z) = (4, -2, 3)

resulting in

22

.

The correct order follows from evaluating the algebraic expression for each set of values according to the order of operations (PEMDAS): exponentiation first, then multiplication, and finally subtraction. This yields values of

-9

,

-5

,

8

, and

22

, which are ordered from least to greatest.

Step-by-Step Solution

1

Evaluate the expression

A = x - y \times z^2

for the set

(x, y, z) = (-5, 1, 2)

.

A = -9

According to the order of operations, evaluate the exponent first:

2^2 = 4

. Then perform multiplication:

1 \times 4 = 4

. Finally, perform subtraction:

-5 - 4 = -9

.

2

Evaluate the expression

A = x - y \times z^2

for the set

(x, y, z) = (-2, 3, -1)

.

A = -5

Evaluate the exponent first:

(-1)^2 = 1

. Next, multiply:

3 \times 1 = 3

. Finally, subtract:

-2 - 3 = -5

.

3

Evaluate the expression

A = x - y \times z^2

for the set

(x, y, z) = (8, 4, 0)

.

A = 8

Evaluate the exponent first:

0^2 = 0

. Next, multiply:

4 \times 0 = 0

. Finally, subtract:

8 - 0 = 8

.

4

Evaluate the expression

A = x - y \times z^2

for the set

(x, y, z) = (4, -2, 3)

.

A = 22

Evaluate the exponent first:

3^2 = 9

. Next, multiply:

-2 \times 9 = -18

. Finally, subtract:

4 - (-18) = 4 + 18 = 22

.

5

Arrange the resulting values from least to greatest.

-9 < -5 < 8 < 22

Comparing the four calculated values shows that

-9

is the least, followed by

-5

, then

8

, and

22

is the greatest.

Key Concept

Order of Operations (PEMDAS)

Alternative Method

To verify the calculations, substitute each set of coordinates carefully, using parentheses around negative values. Evaluate each term step-by-step: first compute

z^2

, then multiply by

y

, and finally subtract this result from

x

.

Estimated Time:1m 30s

Question 114Question →

Four mathematical expressions are shown below:

* Expression 1: $-3^2 - 12 \div 3 \times 2$
* Expression 2: $-| -4 - 2 \times 3 | + (-2)^3$
* Expression 3: $4 - 3(2 - 3^2)$
* Expression 4: $-2^4 \div 8 \times (-1 - 1)^2$

Order the expressions by their simplified numerical values from least to greatest.

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Answer

The correct order of the expressions from least to greatest value is Expression 2, Expression 1, Expression 4, and Expression 3.

Evaluating each expression using the correct order of operations yields: Expression 1 equals

-17

, Expression 2 equals

-18

, Expression 3 equals

25

, and Expression 4 equals

-8

. Arranging these values from least to greatest results in

-18 < -17 < -8 < 25

, corresponding to the sequence: Expression 2, Expression 1, Expression 4, Expression 3.

Step-by-Step Solution

1

Evaluate Expression 1.

-17

Apply the order of operations: first evaluate the exponent

-3^2 = -9

. Next, perform multiplication and division from left to right:

12 \div 3 = 4

, and

4 \times 2 = 8

. Finally, perform the subtraction:

-9 - 8 = -17

.

2

Evaluate Expression 2.

-18

Simplify inside the absolute value first by performing multiplication before subtraction:

2 \times 3 = 6

, and

-4 - 6 = -10

. The absolute value of

-10

is

10

, which becomes

-10

due to the negative sign outside. Next, evaluate the exponent:

(-2)^3 = -8

. Finally, add the terms:

-10 + (-8) = -18

.

3

Evaluate Expression 3.

25

Simplify inside the parentheses first, evaluating the exponent before subtracting:

3^2 = 9

, and

2 - 9 = -7

. Next, perform the multiplication:

3 \times (-7) = -21

. Finally, subtract:

4 - (-21) = 4 + 21 = 25

.

4

Evaluate Expression 4.

-8

Simplify the expression inside the parentheses:

-1 - 1 = -2

. Next, evaluate the exponents:

-2^4 = -16

and

(-2)^2 = 4

. Perform division and multiplication from left to right:

-16 \div 8 = -2

, and

-2 \times 4 = -8

.

5

Compare the simplified values to order them from least to greatest.

-18 < -17 < -8 < 25

Comparing the results shows that the values in ascending order are

-18

(Expression 2),

-17

(Expression 1),

-8

(Expression 4), and

25

(Expression 3).

Key Concept

Order of Operations and Number Properties

Estimated Time:2m 30s

Question 115Question →

Evaluate the four mathematical expressions below, and arrange them in order from least to greatest value.

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Answer

The correct order of expressions from least to greatest is the expression evaluating to

-25

, followed by the expression evaluating to

-8

, then the expression evaluating to

-4

, and finally the expression evaluating to

16

.

Evaluating each expression using the correct order of operations yields

-25

for the expression containing the negative base exponent,

-8

for the expression containing the absolute value,

-4

for the expression containing the cubed term, and

16

for the expression containing the squared term. Ordering these values from least to greatest results in the sequence

-25 < -8 < -4 < 16

.

Step-by-Step Solution

1

Evaluate the first expression:

2 \times (-3)^2 - 4^2 \div 8

.

Value is

16

.

Exponents are calculated first:

(-3)^2 = 9

and

4^2 = 16

. Then, perform multiplication and division:

2 \times 9 = 18

and

16 \div 8 = 2

. Finally, subtract:

18 - 2 = 16

.

2

Evaluate the second expression:

-2^4 + 3 \times (5 - 8)

.

Value is

-25

.

Parentheses are evaluated first:

5 - 8 = -3

. Next, evaluate the exponent:

-2^4 = -16

(since the exponent applies to the base

2

before negation). Perform multiplication:

3 \times (-3) = -9

. Finally, add:

-16 + (-9) = -25

.

3

Evaluate the third expression:

(3 - 5)^3 \div 4 - 2

.

Value is

-4

.

Parentheses are evaluated first:

3 - 5 = -2

. Next, cube the result:

(-2)^3 = -8

. Perform division:

-8 \div 4 = -2

. Finally, subtract:

-2 - 2 = -4

.

4

Evaluate the fourth expression:

-| -6 - 2^2 \times 3 | + 10

.

Value is

-8

.

Within the absolute value bars, evaluate the exponent first:

2^2 = 4

, then multiply:

4 \times 3 = 12

. Subtract to find the absolute value interior:

-6 - 12 = -18

. The absolute value of

-18

is

18

, which is negated to

-18

. Finally, add

10

to get

-18 + 10 = -8

.

5

Order the final values from least to greatest.

-25 < -8 < -4 < 16

.

Comparing the values

-25

,

-8

,

-4

, and

16

on the number line determines their relative size.

Key Concept

Order of operations rules dictate that expressions are evaluated in the sequence of Parentheses/Grouping symbols, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Unary negation is performed after exponentiation unless grouped by parentheses.

Estimated Time:1m 30s

Question 116Question →

What is the correct order of the following numbers from least to greatest?

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Answer

The correct order from least to greatest is:

6.8 \times 10^{-5}

,

7.1 \times 10^{-4}

,

1.5 \times 10^{-3}

, and

2.4 \times 10^{-3}

.

The correct order is determined by first arranging the values by their power of 10 exponent from least to greatest:

-5 < -4 < -3

. This places

6.8 \times 10^{-5}

as the smallest and

7.1 \times 10^{-4}

as the second smallest. For the remaining two values that share the exponent

-3

, we compare their coefficients:

1.5 < 2.4

, placing

1.5 \times 10^{-3}

before

2.4 \times 10^{-3}

.

Step-by-Step Solution

1

Compare the exponents of the base 10 to establish the primary order of magnitude.

The exponents are

-5

,

-4

, and

-3

. Since

-5 < -4 < -3

, we know that any number with an exponent of

-5

is smaller than one with

-4

, which in turn is smaller than one with

-3

.

In scientific notation, a smaller exponent on the power of 10 indicates a smaller overall value because it represents shifting the decimal point further to the left.

2

Place the numbers with unique exponents in order.

The smallest number is

6.8 \times 10^{-5}

, followed by

7.1 \times 10^{-4}

.

These have the smallest exponents (

-5

and

-4

, respectively).

3

Compare the coefficients of the remaining numbers that share the same exponent.

Both

1.5 \times 10^{-3}

and

2.4 \times 10^{-3}

share the exponent

-3

. Comparing their coefficients,

1.5 < 2.4

, so

1.5 \times 10^{-3} < 2.4 \times 10^{-3}

.

When the powers of 10 are identical, the values can be compared directly by their coefficients.

Key Concept

Comparing and ordering numbers written in scientific notation by analyzing their powers of 10 and coefficients.

Question 117Question →

The following sentences are from a paragraph about the use of falconry at airports. Arrange the sentences in the most logical order to create a cohesive paragraph that begins with a clear introduction of the problem and ends with a strong concluding thought.

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Answer

The correct sequence begins with the introduction of the bird strike threat, followed by the introduction of falconry as a solution, then the details of how the raptors deter flocking birds, and ends with the concluding summary sentence.

The correct order establishes a logical progression: first, the paragraph defines the problem of bird strikes; second, it introduces falconry as the solution; third, it describes how the raptors operate; and finally, it concludes by reflecting on the effectiveness of this ancient art for a modern problem.

Step-by-Step Solution

1

Identify the introductory sentence.

The sentence about bird strikes posing a threat to aircraft at busy airports is identified as the opener because it introduces the primary topic without relying on external pronouns or transitional hooks.

An effective introduction must establish the subject and context independently.

2

Determine the logical next sentence by looking for transitional links.

The sentence introducing falconry is placed second because it starts with 'To combat this hazard,' which directly refers back to the threat of bird strikes.

Cohesion is maintained by connecting the solution directly to the defined hazard.

3

Find the detail sentence that follows the solution introduction.

The sentence detailing how the raptors deter flocking birds is placed third because it uses the pointer 'These raptors' to refer back to the 'trained birds of prey'.

Pronoun antecedents must be established in preceding sentences to avoid ambiguity.

4

Identify the concluding sentence.

The sentence starting with 'Ultimately' is placed last because it summarizes the overall paragraph and provides a cohesive conclusion.

Concluding sentences resolve the discussion by connecting the specific solution back to the broader challenge.

Key Concept

Opening and Concluding Sentences

Estimated Time:1m 30s

Question 118Question →

Arrange the following mathematical expressions in order of their simplified values from least to greatest.

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Answer

The correct order of the expressions from least to greatest simplified value is:

-3 - (4 - 2)

, then

-3^2 + 4 \times 2

, then

(-3)^2 - 4 \times 2

, and finally

-(3 - 4 \times 2)

.

Evaluating each expression using the correct order of operations yields

-5

for

-3 - (4 - 2)

,

-1

for

-3^2 + 4 \times 2

,

1

for

(-3)^2 - 4 \times 2

, and

5

for

-(3 - 4 \times 2)

. Ordering these results from least to greatest yields the sequence:

-3 - (4 - 2)

, followed by

-3^2 + 4 \times 2

, then

(-3)^2 - 4 \times 2

, and finally

-(3 - 4 \times 2)

.

Step-by-Step Solution

1

Simplify the expression

-3 - (4 - 2)

.

-3 - (4 - 2) = -3 - 2 = -5

Perform the operation inside the parentheses first, then subtract the result from the leading term.

2

Simplify the expression

-3^2 + 4 \times 2

.

-3^2 + 4 \times 2 = -9 + 8 = -1

Evaluate the exponent first, noting that the negative sign is outside the base (

-3^2 = -9

), then multiply, and finally add the terms.

3

Simplify the expression

(-3)^2 - 4 \times 2

.

(-3)^2 - 4 \times 2 = 9 - 8 = 1

Evaluate the term inside parentheses raised to the power first (

(-3)^2 = 9

), then perform multiplication, and subtract.

4

Simplify the expression

-(3 - 4 \times 2)

.

-(3 - 4 \times 2) = -(3 - 8) = -(-5) = 5

Inside the parentheses, perform multiplication before subtraction, then apply the negative sign outside the parentheses.

5

Compare the simplified values of the four expressions.

-5 < -1 < 1 < 5

Arrange the resulting values from least to greatest to determine the correct sequence.

Key Concept

Applying the correct order of operations (PEMDAS/GEMS) to evaluate numerical expressions, specifically distinguishing between

-a^2

and

(-a)^2

.

Question 119Question →

Arrange the following mathematical values in order from least to greatest.

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Answer

The correct order from least to greatest is

4 \times 10^{-2}

,

2^{-3}

,

\sqrt{0.09}

, and

\frac{3^2}{2 \times 10^1}

.

By converting each value to a decimal, we find

4 \times 10^{-2} = 0.04

,

2^{-3} = 0.125

,

\sqrt{0.09} = 0.3

, and

\frac{3^2}{2 \times 10^1} = 0.45

. Ordering these decimals from smallest to largest yields the correct sequence.

Step-by-Step Solution

1

Convert the scientific notation expression

4 \times 10^{-2}

to its decimal form.

4 \times 10^{-2} = 0.04

Multiplying by

10^{-2}

is equivalent to moving the decimal point of the coefficient

4

two places to the left.

2

Convert the negative exponent expression

2^{-3}

to a fraction and then to a decimal.

2^{-3} = \frac{1}{2^3} = \frac{1}{8} = 0.125

A negative exponent indicates the reciprocal of the base raised to the positive power.

3

Evaluate the square root of the decimal

\sqrt{0.09}

.

\sqrt{0.09} = 0.3

The square root of a decimal less than

1

results in a larger decimal value because

0.3 \times 0.3 = 0.09

.

4

Evaluate the exponential fraction expression

\frac{3^2}{2 \times 10^1}

.

\frac{3^2}{2 \times 10^1} = \frac{9}{20} = 0.45

Simplify the numerator to

9

and the denominator to

20

, then divide to find the decimal equivalent.

5

Compare the evaluated decimal values to determine the correct order from least to greatest.

0.04 < 0.125 < 0.3 < 0.45

, which correspond to

4 \times 10^{-2} < 2^{-3} < \sqrt{0.09} < \frac{3^2}{2 \times 10^1}

.

Comparing decimal place values shows that

0.04

is the smallest and

0.45

is the largest.

Key Concept

Converting and comparing exponential, radical, and scientific notation expressions to a common decimal format.

Question 120Question →

Four students are comparing the portion of their homework assignments they have completed:

* Jamie has completed $35\%$ of his assignment.
* Kayla has completed $\frac{3}{8}$ of her assignment.
* Leo has completed $0.4$ of his assignment.
* Mia has completed $\frac{1}{2}$ of her assignment.

Arrange the students by the portion of their homework completed, from least to greatest.

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Answer

Jamie (

35\%

), Kayla (

\frac{3}{8}

), Leo (

0.4

), and Mia (

\frac{1}{2}

)

Converting all values to decimals yields

35\% = 0.35

,

\frac{3}{8} = 0.375

,

0.4 = 0.40

, and

\frac{1}{2} = 0.50

. Comparing these decimals gives

0.35 < 0.375 < 0.40 < 0.50

, which corresponds to Jamie, Kayla, Leo, and Mia.

Step-by-Step Solution

1

Convert each portion to a decimal value to make them easy to compare.

Jamie:

35\% = 0.35

Kayla:

\frac{3}{8} = 0.375

Leo:

0.4 = 0.40

Mia:

\frac{1}{2} = 0.50

Converting all values to a common format (decimals) allows for direct numerical comparison.

2

Compare the decimal values from least to greatest.

0.35 < 0.375 < 0.40 < 0.50

This establishes the relative sizes of the portions.

3

Match each decimal back to the student and order them.

Jamie (

35\%

), Kayla (

\frac{3}{8}

), Leo (

0.4

), Mia (

\frac{1}{2}

)

This provides the final ordered sequence.

Key Concept

Converting fractions, decimals, and percents to a single format to compare their values.

Answer

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