Pre-Algebra Practice Questions — ACT

Question 1Question →

A library display shelf has space for $5$ distinct books aligned in a row. The librarian has $3$ different science fiction books and $2$ different biography books. How many different row arrangements of these $5$ books are possible if the $2$ biography books must not be placed next to each other?

Show answer & explanation

Answer: 72

Answer

72 different row arrangements are possible.

To find the number of arrangements where the

2

biography books are not adjacent, subtract the number of arrangements where they are adjacent (

4! \times 2! = 48

) from the total possible arrangements (

5! = 120

), resulting in

120 - 48 = 72

.

Step-by-Step Solution

1

Calculate the total number of unrestricted arrangements of all 5 distinct books.

5! = 5 \times 4 \times 3 \times 2 \times 1 = 120

There are

5

distinct books, so there are

5!

total ways to arrange them in a line.

2

Calculate the number of arrangements in which the 2 biography books are placed next to each other.

4! \times 2! = 24 \times 2 = 48

Treat the

2

biography books as a single combined unit. This leaves

4

units to arrange (

3

science fiction books plus

1

biography unit), which can be ordered in

4! = 24

ways. Inside the unit, the

2

biography books can be arranged in

2! = 2

ways.

3

Subtract the number of adjacent biography arrangements from the total number of arrangements.

120 - 48 = 72

Subtracting the restricted outcomes (biography books adjacent) from the total possible outcomes gives the number of valid arrangements (biography books separated).

Key Concept

Permutations and Counting Methods with Complementary Counting

Question 2Question →

Four different saline solutions contain varying concentrations of dissolved salt: Solution P contains $\frac{7}{20}$ , Solution Q contains $36.5\%$ , Solution R contains $\frac{3}{8}$ , and Solution S contains $0.38$ salt concentration. Place the four solutions in order from least concentration to greatest concentration.

Drag items to arrange them in the correct order

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Answer

The correct order from least to greatest concentration is Solution P (7/20), Solution Q (36.5%), Solution R (3/8), and Solution S (0.38).

Converting each quantity to a decimal gives Solution P = 0.350, Solution Q = 0.365, Solution R = 0.375, and Solution S = 0.380. Ordering these decimal values from smallest to largest yields Solution P, Solution Q, Solution R, and Solution S.

Step-by-Step Solution

1

Convert all concentration values into equivalent decimal representations.

Solution P:

\frac{7}{20} = 0.350

; Solution Q:

36.5\% = 0.365

; Solution R:

\frac{3}{8} = 0.375

; Solution S:

0.38 = 0.380

.

Converting all numbers to decimals allows for direct numerical comparison.

2

Compare the decimal values digit by digit from left to right.

0.350 < 0.365 < 0.375 < 0.380

Comparing the hundredths and thousandths places establishes the precise order.

3

Map the ordered decimal values back to their corresponding original solutions.

Solution P (

0.350

), Solution Q (

0.365

), Solution R (

0.375

), Solution S (

0.380

).

To complete the sequence from least to greatest.

Key Concept

Converting fractions, decimals, and percentages to a common numerical form (decimals) to compare and order them.

Question 3Question →

Arrange the following mathematical expressions in order from least to greatest value.

Drag items to arrange them in the correct order

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Answer

The correct order of the expressions from least to greatest is

2^3 \times 5^2

,

4 \times 10^2

,

\sqrt{2.5 \times 10^5}

, and

0.06 \times 10^4

.

Evaluating each term yields

2^3 \times 5^2 = 200

,

4 \times 10^2 = 400

,

\sqrt{2.5 \times 10^5} = 500

, and

0.06 \times 10^4 = 600

. Comparing these simplified values shows that

200 < 400 < 500 < 600

, confirming the correct sequence.

Step-by-Step Solution

1

Evaluate the expression with bases 2 and 5.

2^3 \times 5^2 = 8 \times 25 = 200

First compute the exponential terms and then multiply the resulting values.

2

Evaluate the expression in standard scientific notation.

4 \times 10^2 = 4 \times 100 = 400

Multiply 4 by the value of

10^2

.

3

Evaluate the square root expression.

\sqrt{2.5 \times 10^5} = \sqrt{25 \times 10^4} = 5 \times 10^2 = 500

Adjust the expression under the square root to make it easier to simplify:

2.5 \times 10^5 = 25 \times 10^4

. The square root of 25 is 5, and the square root of

10^4

is

10^2

.

4

Evaluate the non-standard power of ten expression.

0.06 \times 10^4 = 0.06 \times 10,000 = 600

Multiply the decimal coefficient by the power of ten by shifting the decimal point four places to the right.

5

Compare and order all the simplified values.

200 < 400 < 500 < 600

Sort the values from least to greatest to determine the correct order.

Key Concept

Evaluating expressions involving powers, roots, and scientific notation to compare their values.

Question 4Question →

Which of the following is equivalent to the expression $\frac{(5^3)^4}{5^2}$ ?

5^5

5^6

5^{10}

5^{14}

5^{24}

Show answer & explanation

Answer: $5^{10}$

Answer

The correct simplified expression is

5^{10}

.

The expression can be simplified using basic exponent laws. First, using the power of a power rule, the numerator

(5^3)^4

simplifies to

5^{3 \times 4} = 5^{12}

. Then, using the quotient rule, dividing

5^{12}

by

5^2

simplifies to

5^{12 - 2} = 5^{10}

. This corresponds to the option containing

5^{10}

.

Step-by-Step Solution

1

Simplify the numerator using the power of a power rule:

(a^m)^n = a^{m \times n}

.

(5^3)^4 = 5^{3 \times 4} = 5^{12}

When raising a power to another power, multiply the exponents.

2

Simplify the fraction using the quotient rule:

\frac{a^m}{a^n} = a^{m - n}

.

\frac{5^{12}}{5^2} = 5^{12 - 2} = 5^{10}

When dividing exponential expressions with the same base, subtract the exponent of the denominator from the exponent of the numerator.

Key Concept

Applying exponent rules (power of a power rule and quotient rule) to simplify exponential expressions.

Alternative Method

Alternatively, you can write out the terms as repeated multiplication:

(5^3)^4 = 5^3 \times 5^3 \times 5^3 \times 5^3 = 5^{12}

, and then divide by

5^2

by cancelling out two factors of 5, which leaves ten factors of 5, or

5^{10}

.

Estimated Time:45s

Question 5Question →

A recipe for fruit punch calls for orange juice and cranberry juice to be mixed in a ratio of $3:4$ . If a chef uses $12$ cups of orange juice, how many cups of cranberry juice should be used?

4

9

13

16

28

Show answer & explanation

Answer: 16

Answer

16 cups

Since the ratio of orange juice to cranberry juice is 3 to 4, we can set up the proportion 3/4 = 12/x, where x is the amount of cranberry juice. Solving for x yields 16 cups.

Step-by-Step Solution

1

Set up a proportion based on the ratio of orange juice to cranberry juice.

\frac{3}{4} = \frac{12}{x}

where

x

represents the number of cups of cranberry juice.

The ratio of orange juice to cranberry juice is constant at 3 to 4.

2

Solve the proportion for

x

by cross-multiplying.

3x = 12 \times 4

3x = 48

Cross-multiplication is a standard method to solve proportions.

3

Divide both sides by 3 to isolate

x

.

x = 16

Isolating the variable gives the required amount of cranberry juice.

Key Concept

Solving a part-to-part ratio problem using proportions.

Question 6Question →

What is the value of the expression $\frac{-2^4 - 6 \times |2 - 2 \times 3|}{2 - (-2)^3}$ ?

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Answer: -4

Answer

The value of the expression is

-4

.

Evaluating the expression step-by-step using order of operations (PEMDAS) yields

-4

. First, simplifying the absolute value

|2 - 2 \times 3|

gives

4

. Next, the exponent

-2^4

evaluates to

-16

, and the exponent

(-2)^3

in the denominator evaluates to

-8

. The numerator simplifies to

-16 - 6(4) = -40

, and the denominator simplifies to

2 - (-8) = 10

. Finally, division gives

-40 \div 10 = -4

.

Step-by-Step Solution

1

Simplify the absolute value expression in the numerator

|2 - 2 \times 3| = |-4| = 4

By the order of operations, multiplication must be performed before subtraction inside the grouping symbols (absolute value).

2

Evaluate the exponential terms in the numerator and denominator

-2^4 = -16

and

(-2)^3 = -8

In the term

-2^4

, the negative sign is not grouped with the base, so it evaluates to

-(2 \times 2 \times 2 \times 2) = -16

. In

(-2)^3

, the base is negative, evaluating to

(-2) \times (-2) \times (-2) = -8

.

3

Calculate the final values of the numerator and the denominator

Numerator =

-40

, Denominator =

10

For the numerator, perform the multiplication before the subtraction:

-16 - (6 \times 4) = -16 - 24 = -40

. For the denominator, subtract the negative number:

2 - (-8) = 2 + 8 = 10

.

4

Divide the numerator by the denominator

-4

The quotient of

-40

and

10

is

-4

.

Key Concept

Order of operations (PEMDAS) containing exponents, absolute values, and signed numbers

Estimated Time:1m 15s

Question 7Question →

If $a$ , $b$ , and $c$ are positive real numbers such that $a(b + c) = 20$ , $b(c + a) = 13$ , and $c(a + b) = 25$ , what is the value of the product $abc$ ?

Show answer & explanation

Answer: 24

Answer

The product

abc

is equal to 24.

By applying the distributive property, we expand the system to

ab + ac = 20

,

ab + bc = 13

, and

bc + ac = 25

. Adding these yields

2(ab + bc + ac) = 58

, so

ab + bc + ac = 29

. We isolate the individual products:

bc = 9

,

ac = 16

, and

ab = 4

. Multiplying these gives

(abc)^2 = 4 \times 9 \times 16 = 576

. Since the variables are positive,

abc = \sqrt{576} = 24

.

Step-by-Step Solution

1

Expand the equations using the distributive property.

ab + ac = 20

,

ab + bc = 13

, and

bc + ac = 25

This allows us to work with the pairwise products

ab

,

bc

, and

ac

directly.

2

Sum the three equations and divide by 2.

ab + bc + ac = 29

By adding the equations, each pairwise product appears twice:

(ab + ac) + (ab + bc) + (bc + ac) = 2(ab + bc + ac) = 58

. Dividing by 2 gives their sum.

3

Solve for each pairwise product by subtracting the original equations from the sum.

bc = 9

,

ac = 16

, and

ab = 4

Subtracting

ab + ac = 20

from

ab + bc + ac = 29

isolates

bc = 9

. Similarly, subtracting

ab + bc = 13

isolates

ac = 16

, and subtracting

bc + ac = 25

isolates

ab = 4

.

4

Multiply the pairwise products and take the square root.

(abc)^2 = 576 \implies abc = 24

Multiplying

(ab)(bc)(ac)

gives

(abc)^2

. Since

a

,

b

, and

c

are positive, their product

abc

must also be positive, so we take the positive square root of

576

, which is

24

.

Key Concept

Distributive property of multiplication over addition, and properties of equality in systems of equations.

Question 8Question →

For all non-zero real numbers $a$ , $b$ , and $c$ , which of the following is equivalent to the expression $\frac{a(b - c) - b(a - c)}{c}$ ?

b - a

a - b

a + b

ab - a - b

\frac{2ab}{c} - a + b

Show answer & explanation

Answer: $b - a$

Answer

b - a

Applying the distributive property to the numerator yields

ab - ac - ba + bc

. Since multiplication is commutative,

ba = ab

, allowing us to cancel the

ab

and

-ba

terms. This leaves

bc - ac

in the numerator. Factoring out

c

gives

c(b - a)

. Dividing the entire numerator by the denominator

c

simplifies the expression to

b - a

.

Step-by-Step Solution

1

Apply the distributive property to expand the terms in the numerator.

a(b - c) - b(a - c) = ab - ac - ba + bc

Distribute

a

to both

b

and

-c

, and distribute

-b

to both

a

and

-c

. Remember that multiplying two negative numbers yields a positive result, so

-b \times -c = bc

.

2

Combine like terms in the expanded numerator.

ab - ac - ab + bc = bc - ac

Since multiplication is commutative,

ba = ab

. Thus, the terms

ab

and

-ab

cancel each other out to zero.

3

Factor out the common term

c

from the numerator and divide by the denominator

c

.

\frac{c(b - a)}{c} = b - a

Factoring out

c

from

bc - ac

gives

c(b - a)

. Dividing by

c

cancels the common factor in the numerator and denominator, leaving the simplified expression.

Key Concept

Order of Operations and Number Properties

Question 9Question →

For all real numbers $a$ and $b$ , a custom operation $\ast$ is defined by $a \ast b = a^2 - 2b$ . What is the value of the expression $2 \ast (3 \ast 4)$ ?

2

-4

8

-52

-3

Show answer & explanation

Answer: 2

Answer

2

Evaluating the parentheses first gives

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

. Substituting this back into the expression yields

2 \ast 1

. Evaluating this gives

2^2 - 2(1) = 4 - 2 = 2

, which is the correct value.

Step-by-Step Solution

1

Evaluate the expression inside the parentheses:

3 \ast 4

.

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

According to the order of operations, expressions inside parentheses must be evaluated first.

2

Substitute the result back into the main expression.

2 \ast (3 \ast 4) = 2 \ast 1

The evaluated value of the parentheses is substituted back to continue solving.

3

Evaluate

2 \ast 1

using the operation definition.

2 \ast 1 = 2^2 - 2(1) = 4 - 2 = 2

Applying the definition of the custom operation to the remaining terms yields the final answer.

Key Concept

Order of Operations and Custom Arithmetic Operations

Estimated Time:1m 0s

Question 10Question →

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

Drag items to arrange them in the correct order

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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 11Question →

What is the value of the expression $5 - 3 \times (4 - 6)^2$ ?

8

17

-7

-31

65

Show answer & explanation

Answer: -7

Answer

The correct value is

-7

.

The correct value is

-7

. Following the order of operations (PEMDAS), we evaluate the terms inside the parentheses first:

4 - 6 = -2

. Next, we evaluate the exponent:

(-2)^2 = 4

. After that, we perform the multiplication:

3 \times 4 = 12

. Finally, we perform the subtraction:

5 - 12 = -7

.

Step-by-Step Solution

1

Evaluate the subtraction inside the parentheses.

4 - 6 = -2

Operations inside parentheses must be performed first.

2

Evaluate the exponent.

(-2)^2 = 4

Exponents are evaluated after parentheses and before multiplication or division.

3

Perform the multiplication.

3 \times 4 = 12

Multiplication must be performed before subtraction.

4

Perform the final subtraction.

5 - 12 = -7

Subtraction is performed last in this expression.

Key Concept

Order of Operations

Estimated Time:45s

Question 12Question →

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

Drag items to arrange them in the correct order

Show answer & explanation

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 13Question →

What is the value of the expression $|-8| - |6 - 11|$ ?

-13

-9

-3

3

13

Show answer & explanation

Answer: 3

Answer

3

To find the value of the expression, first simplify the expression inside the absolute value bars:

6 - 11 = -5

. Next, apply the definition of absolute value, which makes any value non-negative:

|-8| = 8

and

|-5| = 5

. Finally, subtract the two values:

8 - 5 = 3

.

Step-by-Step Solution

1

Evaluate the subtraction inside the second absolute value grouping.

6 - 11 = -5

Operations inside grouping symbols, such as parentheses or absolute value bars, must be performed first.

2

Apply the absolute value operation to both terms.

|-8| = 8

and

|-5| = 5

The absolute value of a number is its distance from zero, which is always non-negative.

3

Substitute the evaluated absolute values back into the expression and subtract.

8 - 5 = 3

Complete the final subtraction operation as indicated by the expression.

Key Concept

Absolute value acts as a grouping symbol in the order of operations, and the absolute value of any real number is non-negative.

Estimated Time:45s

Question 14Question →

What is the value of the expression $\frac{3 \times 4^2 - 2 \times (5 - 8)^2}{2 - (-2)^3}$ ?

Show answer & explanation

Answer: 3

Answer

The value of the expression is 3.

Following the standard order of operations (PEMDAS), we first calculate the expression inside the parentheses:

5 - 8 = -3

. We then evaluate the exponents:

4^2 = 16

,

(-3)^2 = 9

, and

(-2)^3 = -8

. Substituting these back into the expression, the numerator becomes

3(16) - 2(9) = 48 - 18 = 30

, and the denominator becomes

2 - (-8) = 2 + 8 = 10

. Dividing the numerator by the denominator gives

\frac{30}{10} = 3

.

Step-by-Step Solution

1

Simplify the grouping inside the parentheses

5 - 8 = -3

According to the order of operations, terms inside grouping symbols must be evaluated first.

2

Evaluate the exponential terms

4^2 = 16

,

(-3)^2 = 9

, and

(-2)^3 = -8

Exponents are evaluated next after parentheses.

3

Perform multiplication in the numerator

3 \times 16 = 48

and

2 \times 9 = 18

Multiplication has priority over subtraction.

4

Simplify the numerator and the denominator by performing subtraction and addition

Numerator:

48 - 18 = 30

; Denominator:

2 - (-8) = 10

Addition and subtraction are performed next from left to right.

5

Divide the simplified numerator by the simplified denominator

\frac{30}{10} = 3

Perform the final division to find the value of the fraction.

Key Concept

Order of Operations

Question 15Question →

What is the value of the expression $\frac{18 - 3 \times 4}{|-2| + 1}$ ?

Show answer & explanation

Answer: 2

Answer

The value of the expression is 2.

To evaluate the expression, we simplify the numerator and denominator separately first. In the numerator, we perform the multiplication before the subtraction:

18 - 3 \times 4 = 18 - 12 = 6

. In the denominator, we evaluate the absolute value and then add:

|-2| + 1 = 2 + 1 = 3

. Finally, we divide the simplified numerator by the simplified denominator:

\frac{6}{3} = 2

.

Step-by-Step Solution

1

Evaluate the multiplication in the numerator.

3 \times 4 = 12

, leaving the numerator as

18 - 12

.

According to the order of operations, multiplication must be performed before subtraction.

2

Subtract the values in the numerator.

18 - 12 = 6

.

Simplify the numerator expression completely.

3

Evaluate the absolute value in the denominator.

|-2| = 2

, leaving the denominator as

2 + 1

.

Absolute value represents the non-negative distance from zero, so

|-2| = 2

.

4

Add the values in the denominator.

2 + 1 = 3

.

Simplify the denominator expression completely.

5

Divide the numerator by the denominator.

\frac{6}{3} = 2

.

Compute the final fraction value.

Key Concept

Evaluating expressions using the order of operations (PEMDAS) and absolute value properties.

Question 16Question →

What is the value of the expression $24 - 3 \times (8 - 6)^3$ ?

0

6

168

-192

-864

Show answer & explanation

Answer: $0$

Answer

The value of the expression is

0

.

Applying the standard order of operations (PEMDAS), the parentheses are simplified first to get

2

, then the exponent is evaluated to get

2^3 = 8

. Next, the multiplication is performed to get

3 \times 8 = 24

. Finally, subtraction gives

24 - 24 = 0

.

Step-by-Step Solution

1

Simplify the expression inside the parentheses.

8 - 6 = 2

According to the order of operations (PEMDAS), operations inside parentheses must be performed first.

2

Evaluate the exponent.

2^3 = 8

Exponents are evaluated after parentheses and before multiplication or division.

3

Perform the multiplication.

3 \times 8 = 24

Multiplication is performed after exponents and before subtraction.

4

Perform the subtraction.

24 - 24 = 0

Subtraction is the final step in this expression.

Key Concept

Order of Operations

Estimated Time:45s

Question 17Question →

If $x = -3$ , what is the value of the expression $2x^2 - 4(x - 1) + |x - 5|$ ?

6

10

39

42

60

Show answer & explanation

Answer: 42

Answer

42

The correct answer is 42. Following the order of operations, we first substitute

x = -3

to get

2(-3)^2 - 4(-3 - 1) + |-3 - 5|

. First, evaluate grouping terms:

(-3 - 1) = -4

and

|-3 - 5| = |-8| = 8

. Next, evaluate exponents:

(-3)^2 = 9

. Then, perform multiplications:

2 \times 9 = 18

and

-4 \times -4 = 16

. Finally, perform addition from left to right:

18 + 16 + 8 = 42

.

Step-by-Step Solution

1

Substitute

x = -3

into the given expression.

2(-3)^2 - 4((-3) - 1) + |(-3) - 5|

This sets up the expression for numerical evaluation.

2

Evaluate the terms inside the parentheses and absolute value bars.

Parentheses:

-3 - 1 = -4

; Absolute value:

|-3 - 5| = |-8| = 8

According to the order of operations (PEMDAS), operations inside grouping symbols must be performed first.

3

Evaluate the exponential term.

(-3)^2 = 9

Exponents must be calculated after grouping symbols and before multiplication.

4

Perform the multiplications from left to right.

2(9) = 18

and

-4(-4) = 16

Multiplication takes precedence over addition and subtraction.

5

Add the remaining values.

18 + 16 + 8 = 42

Addition and subtraction are performed from left to right to get the final result.

Key Concept

Evaluating algebraic expressions containing variables, exponents, parentheses, and absolute values using the standard order of operations (PEMDAS).

Estimated Time:1m 15s

Question 18Question →

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

Drag items to arrange them in the correct order

Show answer & explanation

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 19Question →

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.

Drag items to arrange them in the correct order

Show answer & explanation

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 20Question →

What is the value of the expression $4 \times (1 - 3)^2 - 10 \div 2$ ?

-21

3

11

27

59

Show answer & explanation

Answer: $11$

Answer

11

Evaluating the expression following PEMDAS rules: first compute inside the parentheses to get

-2

, then square it to get

4

. Next, perform multiplication (

4 \times 4 = 16

) and division (

10 \div 2 = 5

). Finally, subtract the division result from the multiplication result (

16 - 5 = 11

).

Step-by-Step Solution

1

Evaluate the expression inside the parentheses.

1 - 3 = -2

According to the order of operations (PEMDAS), parentheses must be evaluated first.

2

Evaluate the exponent.

(-2)^2 = 4

Exponents are evaluated after parentheses and before multiplication or division.

3

Perform the multiplication and division from left to right.

4 \times 4 = 16

and

10 \div 2 = 5

Multiplication and division have equal priority and are performed from left to right.

4

Perform the subtraction.

16 - 5 = 11

Addition and subtraction have the lowest priority and are performed last.

Key Concept

Order of Operations (PEMDAS)

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