Tüm alıştırma soruları

Soru 5181Soru →

A zip line is being installed from an observation deck down to a ground anchor. The observation deck is $45$ yards directly above the base of the structure. The ground anchor is secured on flat ground $24$ yards horizontally away from the base. Assuming the zip line is pulled perfectly straight, what is the required length of the zip line cable in feet?

51

69

153

207

Cevabı ve açıklamayı göster

Cevap: 153

Cevap

153

The correct answer accurately calculates the diagonal length using the Pythagorean theorem (

45^2 + 24^2 = c^2

, so

c = 51

yards) and then converts that distance into feet by multiplying by 3 (

51 \times 3 = 153

feet).

Adım Adım Çözüm

1

Apply the Pythagorean theorem (

a^2 + b^2 = c^2

) to find the diagonal length of the zip line in yards.

24^2 + 45^2 = 576 + 2025 = 2601

square yards.

The vertical height and horizontal distance form a right triangle with the zip line acting as the hypotenuse.

2

Calculate the square root of the sum to find the actual length.

\sqrt{2601} = 51

yards.

This isolates

c

(the hypotenuse) to give the actual length of the zip line in yards.

3

Convert the length from yards to feet.

51 \times 3 = 153

feet.

The question specifically asks for the length in feet, and there are

3

feet in one yard.

Anahtar Kavram

Using the Pythagorean theorem to find a hypotenuse and converting units of measurement.

Soru 5182Soru →

Field maintenance regulations for tactical reconnaissance drones dictate specific inspection criteria for composite propeller leading edges. Technicians must inspect the protective erosion tape for signs of peeling or sealant failure. When peeling extends greater than 0.75 inches from the blade tip, or whenever moisture breaches the underlying polyurethane sealant, maintenance crews must immediately replace the entire propeller assembly. In contrast, shallow surface abrasions smaller than 0.25 inches can be repaired locally with 400-grit abrasive cloth while the aircraft remains in service. Complete ultrasonic structural scans are reserved for aircraft reaching 150 flight hours or those subjected to rough landings.

According to the passage, under which condition must technicians immediately replace the entire propeller assembly?

Surface abrasions on the leading edge measure smaller than 0.25 inches

Moisture breaches the underlying polyurethane sealant

Erosion tape peeling extends 0.25 inches from the blade tip

The aircraft accumulates 150 flight hours under normal operations

Cevabı ve açıklamayı göster

Cevap: Moisture breaches the underlying polyurethane sealant

Cevap

Moisture breaches the underlying polyurethane sealant

The passage explicitly identifies two distinct conditions that mandate immediate replacement of the propeller assembly: erosion tape peeling greater than 0.75 inches from the tip, or moisture penetrating the underlying polyurethane sealant. The option specifying moisture breaching the sealant directly reflects this stated fact.

Adım Adım Çözüm

1

Identify the key question focus

The prompt asks for the explicit condition requiring immediate replacement of the entire propeller assembly.

Direct fact retrieval requires scanning for the specific stated trigger for mandatory part replacement.

2

Scan the passage for the replacement criteria

The text states that replacement is mandatory if tape peeling exceeds 0.75 inches or if moisture breaches the underlying polyurethane sealant.

Direct retrieval relies strictly on verbatim factual conditions provided in the text.

3

Match findings to the correct option

The option specifying moisture penetrating the polyurethane sealant matches the stated condition exactly.

Eliminates distractors presenting non-grounding repairs, mismatched numerical thresholds, or routine scan intervals.

Anahtar Kavram

Direct Fact Retrieval

Tahmini Süre:1m 0s

Soru 5183Soru →

Identify the correct step-by-step sequence to fully simplify the given mathematical expression. Order the operations from first to last:

15 - 3 \times (2 + 1)^2

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

The sequence is to evaluate the parentheses first, apply the exponent to that result, perform the multiplication, and finally execute the subtraction.

Following the standard order of operations (PEMDAS), you must first resolve the grouping symbols (parentheses), then evaluate the exponent, perform the multiplication, and finally execute the subtraction.

Adım Adım Çözüm

1

Evaluate the terms inside the parentheses:

(2 + 1)

3

Parentheses (grouping symbols) hold the highest priority in the order of operations.

2

Apply the exponent to the result:

3^2

9

Exponents must be evaluated before multiplication and subtraction.

3

Perform the multiplication:

3 \times 9

27

Multiplication is prioritized before subtraction.

4

Perform the subtraction:

15 - 27

-12

Subtraction is the final remaining operation in the expression.

Anahtar Kavram

Order of Operations

Tahmini Süre:45s

Soru 5184Soru →

Read the following passage:

High-altitude transport aircraft utilize regulated bleed air from turbine engines to maintain positive pressure inside hydraulic fluid reservoirs. At cruising altitudes above 30,000 feet, low ambient atmospheric pressure decreases the boiling point of hydraulic fluid and reduces the inlet pressure delivered to engine-driven pumps. Without artificial reservoir pressurization, fluid vaporization and destructive cavitation occur within pump chambers during operation.

Statement: Without supplemental reservoir pressurization, ambient atmospheric pressure at high cruising altitudes is inadequate to prevent hydraulic fluid from boiling and damaging the pumps.

Doğru

Yanlış

Cevabı ve açıklamayı göster

Cevap: True

Cevap

True: Ambient atmospheric pressure at cruising altitude is insufficient on its own to keep hydraulic fluid below its boiling point and prevent cavitation without bleed-air pressurization.

The conclusion directly follows from combining two core premises in the text: at cruising altitude, ambient pressure lowers the boiling point of fluid, and omitting supplemental pressurization results in vaporization and cavitation.

Adım Adım Çözüm

1

Identify the environmental constraints and operational hazards presented in the text.

Cruising altitudes above 30,000 feet feature low ambient atmospheric pressure, which decreases the hydraulic fluid's boiling point and lowers pump inlet pressure.

Understanding the baseline physical environment is necessary to evaluate why normal atmospheric conditions are insufficient.

2

Determine the role of artificial reservoir pressurization relative to these hazards.

Regulated bleed air provides artificial pressurization to prevent the vaporization and destructive pump cavitation that would otherwise happen.

Establishing the function of the pressurization system clarifies what occurs when it is absent.

3

Synthesize the facts to deduce whether natural ambient pressure alone can maintain safe operating conditions.

Natural atmospheric pressure at high altitudes is demonstrably insufficient, making the evaluated statement strictly true.

Connecting the lack of adequate ambient pressure to the necessity of artificial pressurization validates the logical conclusion.

Anahtar Kavram

Drawing Logical Deductions from Multi-Condition Technical Relationships

Soru 5185Soru →

Early steam locomotives relied on mechanical governors to regulate engine speed under varying track loads. In a typical centrifugal governor, two weighted brass spheres spin on a central spindle driven directly by the locomotive's drive axle. As engine velocity increases, centrifugal force drives the spheres outward and upward against gravity, mechanically linkage-closing the throttle valve to reduce steam intake. Conversely, when the locomotive encounters an incline and slows down, the spheres drop inward, opening the valve to admit more steam. This mechanical feedback loop allowed operators to maintain steady boiler efficiency and prevent catastrophic wheel slippage without requiring continuous manual throttle adjustments.

Which of the following best describes the author's primary purpose in writing this passage?

warn locomotive operators about the dangers of catastrophic wheel slippage on steep inclines

argue that centrifugal governors made steam engines more powerful than modern locomotives

explain how an automatic feedback mechanism regulates locomotive engine speed

criticize early train operators for relying excessively on manual throttle adjustments

Cevabı ve açıklamayı göster

Cevap: explain how an automatic feedback mechanism regulates locomotive engine speed

Cevap

The author's primary purpose is to explain how an automatic feedback mechanism regulates locomotive engine speed.

The author systematically outlines the physical components of the centrifugal governor and explains the mechanical process through which rotating spheres adjust the throttle valve during speed changes. Because the entire passage is structured around clarifying how this self-regulating loop operates, the primary intent is to explain the functioning of this automated speed-regulation mechanism.

Adım Adım Çözüm

1

Analyze the main focus and structure of the passage

The text describes the physical components of a centrifugal governor (spindle, brass spheres, throttle linkage) and traces the physical cause-and-effect cycle that adjusts steam intake based on travel speed.

Examining the dominant content and logical organization reveals why the author composed the paragraph.

2

Determine the author's tone and rhetorical function

The passage maintains a neutral, expository tone aimed at describing how a device functions rather than persuading, warning, or criticizing.

Recognizing the expository nature of the writing distinguishes purely explanatory passages from argumentative or cautionary texts.

3

Synthesize the overarching communicative goal

The author's central aim is to inform the reader about how an automated mechanical feedback loop controls steam engine velocity under changing operating conditions.

The correct response must reflect the passage as a cohesive whole rather than an isolated detail or an unsupported inference.

Anahtar Kavram

Inferring author purpose through synthesis of main idea, rhetorical structure, and objective tone.

Tahmini Süre:1m 0s

Soru 5186Soru →

Read the following technical maintenance passage. Based on the synonymous context clue in the text, determine the definition of the target term to complete the statement below.

"During preventative maintenance on hydraulic braking lines, ground crews must ensure the fluid remains limpid—that is, completely clear and transparent—before pressurizing the reservoir. Any suspended sediment or particulate matter can obstruct micro-valves, leading to erratic pressure spikes and potential actuator failure during high-speed taxiing."

Aşağıdaki boşlukları doldurun

According to the context clues in the passage, the word 'limpid' means .

Cevabı ve açıklamayı göster

Cevap

clear (or transparent)

The passage provides a direct synonymous context clue set off by dashes and introduced by the signal phrase 'that is'. The words immediately following ('completely clear and transparent') directly define 'limpid'. Therefore, filling the blank with 'clear' or 'transparent' accurately completes the sentence.

Adım Adım Çözüm

1

Locate the target vocabulary word 'limpid' within the passage.

The word appears in the phrase: 'ensure the fluid remains limpid—that is, completely clear and transparent—before pressurizing the reservoir.'

Identifying the target word's position in the sentence reveals the surrounding punctuation and signaling phrases.

2

Identify the punctuation and signaling phrase that provides a restatement clue.

The em-dashes and the signal phrase 'that is' directly introduce an explanatory restatement of 'limpid'.

In reading comprehension passages, the phrase 'that is' offset by dashes serves as a direct synonymous context clue defining the preceding term.

3

Extract the synonym to complete the blank in the statement.

The clarifying phrase 'completely clear and transparent' supplies the exact meaning, identifying 'clear' or 'transparent' as the correct response.

The author provides the exact meaning within the context of the sentence without requiring outside technical knowledge.

Anahtar Kavram

Identifying direct restatement and appositive synonymous context clues indicated by signal phrases such as 'that is'.

Soru 5187Soru →

What is the minimum integer value of $x$ that satisfies the inequality shown below?

3(2x - 5) - 8x \leq 4(x - 1) - 23

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

2

Solving the inequality yields

x \geq 2

. Since

x

must be greater than or equal to 2, the minimum integer value that makes this statement true is exactly 2.

Adım Adım Çözüm

1

Distribute the constants into the parentheses on both sides of the inequality.

6x - 15 - 8x \leq 4x - 4 - 23

This removes the parentheses, allowing us to combine like terms.

2

Combine the like terms on each side of the inequality.

-2x - 15 \leq 4x - 27

Simplifying each side makes it easier to isolate the variable.

3

Subtract

4x

from both sides to group the variable terms on the left side.

-6x - 15 \leq -27

We need all terms with

x

on one side of the inequality to solve for it.

4

Add

15

to both sides to isolate the term with the variable.

-6x \leq -12

This leaves only the

-6x

term on the left side.

5

Divide both sides by

-6

, remembering to flip the inequality symbol.

x \geq 2

Dividing or multiplying an inequality by a negative number reverses the direction of the inequality sign.

6

Determine the minimum integer value that satisfies the simplified inequality.

2

The inequality states that

x

must be greater than or equal to

2

. The smallest integer that meets this condition is

2

.

Anahtar Kavram

Solving linear inequalities requiring the distributive property, combining like terms, and reversing the inequality symbol when dividing by a negative coefficient.

Soru 5188Soru →

Given the variable assignments $x = -4$ , $y = 2$ , and $z = -6$ , which of the following represents the evaluated result of the mathematical expression below?

x^2 y - \frac{|x - z|}{y + x}

33

-31

-15

31

Cevabı ve açıklamayı göster

Cevap: 33

Cevap

33

Substituting the given values into the expression yields

(-4)^2(2) - \frac{|-4 - (-6)|}{2 + (-4)}

. Squaring

-4

gives

16

, and multiplying by

2

results in

32

. Inside the absolute value,

-4 - (-6)

becomes

-4 + 6 = 2

, so the numerator is

2

. The denominator is

2 + (-4) = -2

. The fraction simplifies to

2 / -2 = -1

. Subtracting this from the first term gives

32 - (-1) = 33

.

Adım Adım Çözüm

1

Substitute the given values for x, y, and z into the expression.

(-4)^2 (2) - \frac{|-4 - (-6)|}{2 + (-4)}

Direct substitution is the first requirement to evaluate an algebraic expression.

2

Evaluate the first term with exponents and multiplication.

(-4)^2 = 16

, then

16 \times 2 = 32

.

Following the order of operations (PEMDAS), exponents are calculated before multiplication.

3

Simplify the expression inside the absolute value in the numerator.

|-4 - (-6)| = |-4 + 6| = |2| = 2

.

Subtracting a negative is equivalent to adding a positive. Absolute value makes the result positive.

4

Simplify the denominator and evaluate the fraction.

Denominator is

2 + (-4) = -2

. The fraction evaluates to

\frac{2}{-2} = -1

.

The operations within the implied grouping of the fraction must be fully resolved before dividing.

5

Subtract the evaluated fraction from the first term.

32 - (-1) = 32 + 1 = 33

.

Final subtraction, ensuring that subtracting a negative value is treated as addition.

Anahtar Kavram

Evaluating Complex Algebraic Expressions with Integers

Soru 5189Soru →

A runner completes $\frac{2}{5}$ of a mile as a warm-up and then runs another $\frac{1}{3}$ of a mile. What is the total distance, in miles, the runner covered?

\frac{11}{15}

\frac{3}{8}

\frac{2}{15}

\frac{1}{5}

Cevabı ve açıklamayı göster

Cevap: $\frac{11}{15}$

Cevap

\frac{11}{15}

To find the total distance, the fractions must be added. Since they have unlike denominators (5 and 3), the lowest common denominator (15) must be found. Converting the fractions yields

\frac{6}{15}

and

\frac{5}{15}

. Adding the numerators provides the final total of

\frac{11}{15}

.

Adım Adım Çözüm

1

Identify the operation needed.

Addition:

\frac{2}{5} + \frac{1}{3}

The question asks for the total distance covered, requiring the sum of the two distinct parts of the run.

2

Find a common denominator for 5 and 3.

The lowest common multiple of 5 and 3 is 15.

Fractions must have a common denominator before they can be added together.

3

Convert both fractions to have a denominator of 15.

\frac{2}{5} = \frac{6}{15}

and

\frac{1}{3} = \frac{5}{15}

To maintain the same value, multiply the numerator and denominator of the first fraction by 3, and the second fraction by 5.

4

Add the numerators and keep the common denominator.

\frac{6}{15} + \frac{5}{15} = \frac{11}{15}

Once denominators are the same, only the numerators are added to find the total.

Anahtar Kavram

Adding Fractions with Unlike Denominators

Soru 5190Soru →

Match each algebraic expression containing numerical coefficients on the left with its fully simplified equivalent form on the right.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

The simplified pairs are:

3y^2 \cdot 2y^4

matches with

6y^6

,

\frac{12y^7}{4y^3}

matches with

3y^4

,

(2y^3)^3

matches with

8y^9

, and

5y^{-2}

matches with

\frac{5}{y^2}

.

Each left-side expression simplifies to exactly one unique form on the right by appropriately applying exponent rules to the variable bases and standard arithmetic operations to the numerical coefficients.

Adım Adım Çözüm

1

Simplify the product expression

3y^2 \cdot 2y^4

.

6y^6

When multiplying terms, handle coefficients and variables separately. Multiply coefficients (

3 \times 2 = 6

) and add exponents for identical bases (

y^{2+4} = y^6

).

2

Simplify the quotient expression

\frac{12y^7}{4y^3}

.

3y^4

Divide the coefficients (

12 \div 4 = 3

) and subtract the bottom exponent from the top exponent (

y^{7-3} = y^4

).

3

Simplify the power expression

(2y^3)^3

.

8y^9

Raise both the coefficient and the variable to the power of 3. Calculate

2^3 = 8

, and multiply the exponents for the variable

3 \times 3 = 9

.

4

Simplify the negative exponent expression

5y^{-2}

.

\frac{5}{y^2}

A negative exponent creates a reciprocal. Since the exponent only applies to

y

, it moves to the denominator while the coefficient 5 stays in the numerator.

Anahtar Kavram

Handling numerical coefficients alongside basic operations with variable exponents.

Soru 5191Soru →

A chef is preparing a special sauce and needs $\frac{7}{8}$ of a cup of broth. They currently have $\frac{1}{4}$ of a cup of broth in a measuring jug. How much more broth must be added to reach the required amount?

\frac{2}{3}

\frac{3}{4}

\frac{5}{8}

1\frac{1}{8}

Cevabı ve açıklamayı göster

Cevap: $\frac{5}{8}$

Cevap

The chef needs

\frac{5}{8}

of a cup of additional broth.

The correct answer is found by taking the total amount needed and subtracting the amount already available. First, convert

\frac{1}{4}

to an equivalent fraction with a denominator of 8, which is

\frac{2}{8}

. Then, subtract

\frac{2}{8}

from

\frac{7}{8}

to get

\frac{5}{8}

.

Adım Adım Çözüm

1

Set up a subtraction problem to find the difference between the total amount of broth needed and the amount currently on hand.

\frac{7}{8} - \frac{1}{4}

Subtracting the available amount from the required total yields the missing remainder.

2

Find a common denominator so the fractions can be subtracted. Multiply the numerator and denominator of the second fraction by 2.

\frac{1}{4} = \frac{1 \times 2}{4 \times 2} = \frac{2}{8}

Fractions must have the same denominator (in this case, 8) before they can be added or subtracted.

3

Subtract the numerators while keeping the common denominator the same.

\frac{7}{8} - \frac{2}{8} = \frac{5}{8}

This calculation provides the final exact fraction of additional broth required.

Anahtar Kavram

Subtracting fractions with unlike denominators by first converting to a common denominator.

Tahmini Süre:45s

Soru 5192Soru →

Compute the value of the algebraic expression $3p^2 - 2pq$ utilizing the parameters $p = -3$ and $q = 5$ .

57

3

-3

111

Cevabı ve açıklamayı göster

Cevap: 57

Cevap

57

The correct answer accurately substitutes the negative and positive integers into the formula and follows PEMDAS. Squaring

-3

yields

9

, which is multiplied by

3

to give

27

. The second term involves multiplying

2

,

-3

, and

5

to yield

-30

. Subtracting

-30

from

27

is equivalent to

27 + 30

, which totals

57

.

Adım Adım Çözüm

1

Substitute the assigned values for p and q into the expression.

3(-3)^2 - 2(-3)(5)

Variable substitution requires replacing every instance of a letter with its numerical value.

2

Evaluate the exponent according to the order of operations.

3(9) - 2(-3)(5)

Exponents must be calculated before multiplication. A negative number squared becomes positive.

3

Perform the multiplications for both terms.

27 - (-30)

Multiply

3 \times 9

to get 27, and multiply

2 \times (-3) \times 5

to get

-30

.

4

Subtract the second term from the first term.

57

Subtracting a negative number is the same as adding its positive equivalent (

27 + 30 = 57

).

Anahtar Kavram

Evaluating Algebraic Expressions

Tahmini Süre:45s

Soru 5193Soru →

In a manufacturing process, the required cooling time in minutes is defined as eight subtracted from twice the sum of the baseline time, $t$ , and four. Which of the following algebraic expressions represents the required cooling time?

2(t + 4) - 8

8 - 2(t + 4)

2t + 4 - 8

2(5t) - 8

Cevabı ve açıklamayı göster

Cevap: $2(t + 4) - 8$

Cevap

The correct algebraic expression is

2(t + 4) - 8

.

The correct expression carefully follows the algebraic translation rules: 'sum' dictates the grouping

(t + 4)

, 'twice' multiplies that entire group by 2 to get

2(t + 4)

, and 'eight subtracted from' places the

- 8

at the end of the expression.

Adım Adım Çözüm

1

Identify and group the addition operation described in the problem.

(t + 4)

The phrase 'the sum of the baseline time,

t

, and four' indicates that these two quantities must be added and kept together as a single unit.

2

Apply the multiplication to the grouped sum.

2(t + 4)

The word 'twice' means multiplying the entire group by 2.

3

Translate the subtraction phrase in the correct order.

2(t + 4) - 8

The phrase 'eight subtracted from' is a turnaround phrase, meaning 8 must be taken away from the grouped expression.

Anahtar Kavram

Translating complex word phrases requires identifying grouping words (like 'sum') to use parentheses, and recognizing turnaround phrases (like 'subtracted from') that reverse the written order in the mathematical expression.

Soru 5194Soru →

What is the final result of the expression $9.25 - 3.4 \times 0.5$ ?

2.925

9.08

7.55

10.95

Cevabı ve açıklamayı göster

Cevap: $7.55$

Cevap

The correct answer is

7.55

.

The expression requires following the proper order of operations, meaning the multiplication must be calculated before the subtraction. First, multiplying

3.4

by

0.5

yields

1.7

. Then, aligning the decimals correctly and subtracting

1.7

from

9.25

provides the final answer of

7.55

.

Adım Adım Çözüm

1

Identify the correct order of operations (PEMDAS/BODMAS) for the expression.

Multiplication must be performed before subtraction.

Standard mathematical rules require evaluating multiplication before addition or subtraction.

2

Multiply

3.4

by

0.5

.

3.4 \times 0.5 = 1.7

Resolving the multiplication part of the expression.

3

Subtract the product from

9.25

, ensuring the decimal points are vertically aligned.

9.25 - 1.70 = 7.55

Completing the final subtraction operation to find the total value.

Anahtar Kavram

Decimal Operations

Tahmini Süre:45s

Soru 5195Soru →

When $p = -2$ and $q = -5$ , which of the following represents the value of $4p - 3q$ ?

7

-7

-23

55

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

7

The correct value is found by substituting

p = -2

and

q = -5

into the expression

4p - 3q

. Applying the order of operations, first multiply to get

-8

and

-15

. Finally, evaluate

-8 - (-15)

, which correctly simplifies to

-8 + 15 = 7

.

Adım Adım Çözüm

1

Substitute

-2

for

p

and

-5

for

q

into the algebraic expression.

4(-2) - 3(-5)

Variables must be replaced with their given numerical values to evaluate the expression.

2

Multiply the terms following the order of operations.

-8 - (-15)

Multiplication must be performed before subtraction.

3

Simplify the resulting subtraction.

-8 + 15 = 7

Subtracting a negative number is mathematically identical to adding a positive number.

Anahtar Kavram

Evaluating Algebraic Expressions

Tahmini Süre:45s

Soru 5196Soru →

A city water purification system processes $8.5 \times 10^5$ liters of water per day. A specialized chemical treatment requires $4.0 \times 10^{-4}$ grams of a purifying agent per liter of water. What is the total mass, in grams, of the purifying agent needed each day? Express the answer in proper scientific notation.

3.4 \times 10^0

3.4 \times 10^{-19}

3.4 \times 10^2

3.4 \times 10^{10}

Cevabı ve açıklamayı göster

Cevap: $3.4 \times 10^2$

Cevap

The correct answer is

3.4 \times 10^2

.

The total mass is calculated by multiplying the daily volume by the required agent per liter:

(8.5 \times 10^5) \times (4.0 \times 10^{-4})

. Multiplying the coefficients gives

34

, and adding the exponents yields

10^1

. The intermediate result,

34 \times 10^1

, normalizes to

3.4 \times 10^2

by shifting the decimal one place to the left and adding

1

to the exponent.

Adım Adım Çözüm

1

Set up the multiplication expression to find the total mass.

(8.5 \times 10^5) \times (4.0 \times 10^{-4})

The total mass is the product of the total daily volume of water and the mass of the purifying agent required per liter.

2

Group the coefficients and the powers of ten.

(8.5 \times 4.0) \times (10^5 \times 10^{-4})

The commutative and associative properties of multiplication allow grouping similar numerical components.

3

Multiply the coefficients and add the exponents.

34 \times 10^1

8.5 \times 4.0 = 34

, and following the product rule of exponents,

10^5 \times 10^{-4} = 10^{5 + (-4)} = 10^1

.

4

Normalize the result into proper scientific notation.

3.4 \times 10^2

The coefficient

34

must be converted to

3.4

(a number between 1 and 10). Because the decimal point is moved one place to the left, the exponent is increased by 1.

Anahtar Kavram

Multiplying Numbers in Scientific Notation and Normalization

Soru 5197Soru →

Arrange the following fractions and mixed numbers in ascending order (from least to greatest).

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

The correct ascending order is

\frac{14}{9}

,

1\frac{5}{8}

,

1\frac{2}{3}

, and

\frac{7}{4}

.

By converting all values to improper fractions and finding a common denominator of 72, the values become

\frac{112}{72}

,

\frac{117}{72}

,

\frac{120}{72}

, and

\frac{126}{72}

. Comparing the numerators allows us to determine the correct ascending order of the original values.

Adım Adım Çözüm

1

Convert all mixed numbers to improper fractions.

1\frac{5}{8} = \frac{13}{8}

and

1\frac{2}{3} = \frac{5}{3}

.

Improper fractions are easier to evaluate when finding a common denominator.

2

Determine the least common multiple (LCM) of the denominators 9, 8, 3, and 4.

The LCM is 72.

A common denominator is required to accurately compare the sizes of fractions.

3

Convert each fraction into an equivalent fraction with a denominator of 72.

\frac{14}{9} = \frac{112}{72}

,

\frac{13}{8} = \frac{117}{72}

,

\frac{5}{3} = \frac{120}{72}

,

\frac{7}{4} = \frac{126}{72}

.

Having a common denominator allows for direct comparison of the numerators.

4

Compare the numerators to order the fractions from least to greatest.

112 < 117 < 120 < 126

, which means

\frac{112}{72} < \frac{117}{72} < \frac{120}{72} < \frac{126}{72}

.

The question specifies arranging the values in ascending order.

5

Map the ordered fractions back to their original forms.

\frac{14}{9}

,

1\frac{5}{8}

,

1\frac{2}{3}

,

\frac{7}{4}

.

To provide the final answer using the exact values given in the prompt.

Anahtar Kavram

Comparing and ordering mixed numbers and improper fractions with unlike denominators by finding a common denominator.

Soru 5198Soru →

Evaluate the following mathematical expression and determine the final integer result.

Aşağıdaki boşlukları doldurun

The value of the expression

\frac{|-15 + 3|}{2} - |-4 \times 2|

is .

Cevabı ve açıklamayı göster

Cevap

The final evaluated result is -2.

The correct result is found by first evaluating the expressions inside the absolute value symbols, which act as grouping symbols. The expression

|-15 + 3|

becomes

|-12|

, which evaluates to

12

. The second part,

|-4 \times 2|

, becomes

|-8|

, which evaluates to

8

. Substituting these simplified values back into the original expression yields

\frac{12}{2} - 8

. Following the order of operations, division is performed first, giving

6

. Finally,

6 - 8

results in

-2

.

Adım Adım Çözüm

1

Evaluate the operations inside the first absolute value bars.

|-15 + 3| = |-12|

Order of operations requires treating absolute value bars as grouping symbols, meaning the arithmetic inside must be completed first.

2

Apply the absolute value to the first result.

|-12| = 12

The absolute value of a number represents its distance from zero, which is always positive.

3

Evaluate the operations inside the second absolute value bars.

|-4 \times 2| = |-8| = 8

Perform the multiplication inside the grouping symbol first, then take the absolute value of the negative product.

4

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

\frac{12}{2} - 8 = 6 - 8 = -2

Perform division before subtraction according to the standard order of operations (PEMDAS).

Anahtar Kavram

Absolute Value and Order of Operations

Soru 5199Soru →

Simplify the following algebraic expression completely:

(5x^{-2}y^4)(-2x^5y^{-1})^2

Which of the following represents the correct equivalent form?

20x^8y^2

20x^5y^5

100x^6y^6

-20x^8y^2

Cevabı ve açıklamayı göster

Cevap: $20x^8y^2$

Cevap

The correctly simplified expression is

20x^8y^2

.

The correct result is achieved by following the order of operations and applying the laws of exponents. First, the exponent of 2 must be distributed to all parts of the second term:

(-2)^2(x^5)^2(y^{-1})^2

, which simplifies to

4x^{10}y^{-2}

. Next, this result is multiplied by the first term:

(5x^{-2}y^4)(4x^{10}y^{-2})

. Multiplying the coefficients gives

5 \times 4 = 20

. Using the product of powers rule, adding the exponents for

x

yields

-2 + 10 = 8

, and for

y

yields

4 + (-2) = 2

. This results in the final expression

20x^8y^2

.

Adım Adım Çözüm

1

Apply the power of a product rule to the second term by squaring the coefficient and each variable.

(-2)^2(x^5)^2(y^{-1})^2 = 4x^{10}y^{-2}

The order of operations (PEMDAS) requires evaluating exponents before performing multiplication.

2

Rewrite the full expression with the expanded term.

(5x^{-2}y^4)(4x^{10}y^{-2})

To clearly identify the coefficients and bases that need to be combined.

3

Multiply the numerical coefficients and apply the product of powers rule to the variables by adding their exponents.

(5 \times 4)x^{-2+10}y^{4+(-2)} = 20x^8y^2

When multiplying exponential terms with the same base, you add the exponents.

Anahtar Kavram

Operations with Variable Exponents

Soru 5200Soru →

Pair each algebraic operation on the left with its correctly simplified expression on the right. Assume all variables represent non-zero real numbers.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

The correct matches require applying the product, quotient, and power rules of exponents systematically to each expression.

Each expression is correctly simplified by identifying the primary exponent rule required (product, quotient, or power) and applying it to both the numerical coefficients and the variables' exponents.

Adım Adım Çözüm

1

Evaluate

(3m^4 n^2)(4m^2 n^5)

using the Product Rule.

12m^6 n^7

Multiply coefficients (3 * 4 = 12) and add exponents of corresponding variables (

m^{4+2}

,

n^{2+5}

).

2

Evaluate

\frac{12m^8 n^3}{4m^2 n^{-4}}

using the Quotient Rule.

3m^6 n^7

Divide coefficients (12 / 4 = 3) and subtract denominator exponents from numerator exponents (

m^{8-2}

,

n^{3-(-4)}

).

3

Evaluate

(2m^3 n^2)^3

using the Power of a Product Rule.

8m^9 n^6

Cube the coefficient (

2^3 = 8

) and multiply inner exponents by the outer exponent (

m^{3 \times 3}

,

n^{2 \times 3}

).

4

Evaluate

\frac{24m^4 n^2}{3m^5 n^{-1}}

using the Quotient Rule and Negative Exponent Rule.

\frac{8n^3}{m}

Divide coefficients (24 / 3 = 8), subtract exponents (

m^{4-5} = m^{-1}

,

n^{2-(-1)} = n^3

), and rewrite the negative exponent in the denominator.

Anahtar Kavram

Applying fundamental exponent laws (product rule, quotient rule, power of a power) and properly handling numerical coefficients separately from variable exponents.

Tahmini Süre:1m 30s

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