Tüm alıştırma soruları

Soru 4861Soru →

In a medical supply drop, the ratio of bandages to antiseptic bottles is $7:3$ . If there are $210$ antiseptic bottles in the drop, how many bandages are included?

Cevabı ve açıklamayı göster

Cevap: 490

Cevap

490

The proportion must match the quantity of bandages to antiseptics. Since the ratio is

7

parts bandages for every

3

parts antiseptics, and the actual number of antiseptics is

210

(

70

times the ratio amount), the number of bandages must also be

70

times its ratio amount. Therefore,

7 \times 70 = 490

bandages.

Adım Adım Çözüm

1

Identify the known ratio and the known quantity.

The ratio of bandages to antiseptics is

7:3

, and there are

210

antiseptics.

To correctly set up a mathematically accurate proportion.

2

Set up the proportion equation.

\frac{7}{3} = \frac{x}{210}

To relate the given simplified ratio to the actual numbers.

3

Solve the proportion for

x

.

x = \frac{7 \times 210}{3} = 7 \times 70 = 490

To find the unknown number of bandages.

Anahtar Kavram

Solving basic proportions from a given part-to-part ratio

Tahmini Süre:45s

Soru 4862Soru →

Read the passage below:

Early military balloon reconnaissance promised commanders an unprecedented view of enemy troop movements, yet commanders often treated the aerial reports with justifiable caution. Ascending in a tethered basket exposed the aeronaut to buffeting winds that blurred optical sightings, while communication via dropped message weights or erratic signal flags routinely caused critical delays. Although the technology appeared groundbreaking on paper, its operational clumsiness frequently turned tactical promises into costly battlefield confusion.

Which of the following best describes the author's attitude toward early military balloon reconnaissance?

Impressed by the groundbreaking clarity of observations made from the basket

Critical of its practical reliability and real-world battlefield execution

Convinced that all military aerial surveillance was fundamentally useless and should have been permanently banned

Indifferent to the physical hazards and signaling difficulties faced by early aeronauts

Cevabı ve açıklamayı göster

Cevap: Critical of its practical reliability and real-world battlefield execution

Cevap

Critical of its practical reliability and real-world battlefield execution

The author maintains a critical attitude toward early military balloon reconnaissance, emphasizing its severe practical shortcomings with phrases like 'justifiable caution,' 'blurred optical sightings,' 'erratic signal flags,' and 'operational clumsiness.' Rather than praising the system, the text emphasizes how its theoretical promise failed in real combat situations.

Adım Adım Çözüm

1

Examine the author's word choices and descriptive phrases regarding the subject.

The author uses evaluative language such as 'justifiable caution,' 'blurred optical sightings,' 'erratic signal flags,' and 'operational clumsiness.'

Tone and attitude are directly conveyed through qualitative adjectives and the emotional weight of descriptive phrasing.

2

Analyze contrast transitions to separate theoretical claims from the author's actual assessment.

The word 'Although' contrasts the idea that the technology was 'groundbreaking on paper' with the reality that it caused 'costly battlefield confusion.'

Contrast structures highlight the author's genuine judgment versus superficial claims.

3

Identify the option that reflects this critical assessment without exaggerating the author's claims.

The author is critical of the practical reliability and real-world execution of the reconnaissance balloons.

This accurately synthesizes the passage's evidence without assuming extreme viewpoints not stated in the text.

Anahtar Kavram

Inferring author tone requires identifying evaluative diction, descriptive modifiers, and contrast transitions that reveal the writer's underlying attitude toward the topic.

Tahmini Süre:1m 0s

Soru 4863Soru →

Find the value of $n$ that makes the following equation true:

\frac{x^9}{x^3} = x^n

Cevabı ve açıklamayı göster

Cevap: 6; six

Cevap

6

According to the quotient rule for exponents, when you divide two exponential terms that have the same base, you subtract the exponent of the denominator from the exponent of the numerator. In this equation, both terms have the base

x

. Subtracting the denominator's exponent (3) from the numerator's exponent (9) gives

9 - 3 = 6

. Therefore, the left side simplifies to

x^6

. Equating this to

x^n

means that

n

must equal 6.

Adım Adım Çözüm

1

Apply the quotient rule for exponents, which states that when dividing terms with identical bases, you subtract the exponent in the denominator from the exponent in the numerator.

x^{9-3} = x^n

To simplify the left side of the equation into a single term with base

x

.

2

Perform the subtraction in the exponent.

x^6 = x^n

Calculating

9 - 3

gives the simplified exponent.

3

Set the resulting exponent equal to

n

.

n = 6

Since the bases are identical (

x

), their exponents must be equal for the equation to hold true.

Anahtar Kavram

Quotient Rule for Exponents

Soru 4864Soru →

Unlike the permanent, well-furnished headquarters established at the main regional airbase, the forward operating outpost was decidedly austere. Personnel stationed at this remote perimeter post slept on canvas cots inside uninsulated tents, lacking running water and standard mess facilities.

As used in the passage, what does the word "austere" most nearly mean?

Luxurious and fully equipped

Simple and lacking comforts

Hostile and dangerous

Severely damaged and broken

Cevabı ve açıklamayı göster

Cevap: Simple and lacking comforts

Cevap

Simple and lacking comforts

The author uses the contrast indicator 'Unlike' to distinguish the forward operating outpost from the 'well-furnished headquarters'. Because the headquarters has extensive amenities while the outpost has uninsulated tents, canvas cots, and no running water, 'austere' means simple, bare, and lacking comforts.

Adım Adım Çözüm

1

Locate the target word and identify any transition or contrast signal words in the surrounding sentence.

The target word 'austere' is introduced by the contrast signal 'Unlike', which compares the forward operating outpost to the main airbase.

Contrast words indicate that the two items being compared have opposite characteristics.

2

Examine the description of the opposing element and supporting details.

The main airbase is described as 'permanent' and 'well-furnished', while the outpost features 'canvas cots', 'uninsulated tents', and lacks 'running water'.

The opposite of 'well-furnished' and having standard amenities indicates a plain, bare, and unadorned environment.

3

Match the inferred antonymous meaning with the correct answer choice.

'Simple and lacking comforts' precisely captures the opposite of 'well-furnished' in the context of living conditions.

This meaning directly fits the context established by the author's contrast.

Anahtar Kavram

Identifying Antonym and Contrast Context Clues

Tahmini Süre:1m 0s

Soru 4865Soru →

A military supply officer is preparing cargo for an allied joint exercise. Match the following US Customary measurements to their approximate Metric equivalents based on standard conversion rates ( $1 \text{ inch} = 2.54 \text{ cm}$ , $1 \text{ kg} \approx 2.2 \text{ lbs}$ , $1 \text{ mile} \approx 1.6 \text{ km}$ , $1 \text{ gallon} \approx 3.78 \text{ L}$ ).

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

The correct pairings are:

10 \text{ inches}

matches with

25.4 \text{ centimeters}

,

5 \text{ miles}

matches with

8 \text{ kilometers}

,

110 \text{ pounds}

matches with

50 \text{ kilograms}

, and

10 \text{ gallons}

matches with

37.8 \text{ liters}

.

Each measurement is paired with its metric equivalent by correctly applying the given conversion factors. Multiplying is used to convert inches to centimeters, miles to kilometers, and gallons to liters. Dividing is required to convert pounds to kilograms.

Adım Adım Çözüm

1

Identify the conversion factors provided in the prompt.

1 \text{ inch} = 2.54 \text{ cm}

,

1 \text{ kg} = 2.2 \text{ lbs}

,

1 \text{ mile} = 1.6 \text{ km}

,

1 \text{ gallon} = 3.78 \text{ L}

These rates determine whether to multiply or divide.

2

Convert the clearance height from inches to centimeters.

10 \times 2.54 = 25.4 \text{ cm}

Multiplying by the factor converts US standard length to metric.

3

Convert the marching distance from miles to kilometers.

5 \times 1.6 = 8 \text{ km}

Multiplying miles by

1.6

yields the correct distance in kilometers.

4

Convert the crate weight from pounds to kilograms.

110 \div 2.2 = 50 \text{ kg}

Since

1 \text{ kg}

is

2.2 \text{ lbs}

, dividing the total pounds by

2.2

gives the number of kilograms.

5

Convert the fuel capacity from gallons to liters.

10 \times 3.78 = 37.8 \text{ L}

Multiplying gallons by the conversion factor

3.78

yields the volume in liters.

Anahtar Kavram

Inter-System Unit Conversions

Soru 4866Soru →

Consider the following linear inequality:

3(x - 2) - 7x \geq 2(x + 3)

Which of the following represents the correct solution set for $x$ ?

x \leq -2

x \geq -2

x \leq 0

x \geq 6

Cevabı ve açıklamayı göster

Cevap: $x \leq -2$

Cevap

The correct solution is

x \leq -2

.

To find the solution, expand the terms to get

3x - 6 - 7x \geq 2x + 6

. Combine the like terms on the left to form

-4x - 6 \geq 2x + 6

. Subtracting

2x

and adding

6

to both sides isolates the variable, resulting in

-6x \geq 12

. Finally, divide by

-6

. Because we are dividing by a negative number, the inequality sign must flip, giving the correct answer of

x \leq -2

.

Adım Adım Çözüm

1

Distribute the constants on both sides of the inequality.

3x - 6 - 7x \geq 2x + 6

Expanding the expressions clears the parentheses and allows us to combine like terms.

2

Combine the like terms on the left side of the inequality.

-4x - 6 \geq 2x + 6

3x - 7x

simplifies to

-4x

.

3

Move all terms containing

x

to one side and constants to the other side.

-6x \geq 12

Subtracting

2x

from both sides gives

-6x - 6 \geq 6

. Then, adding

6

to both sides isolates the variable term.

4

Divide both sides by

-6

and reverse the inequality symbol.

x \leq -2

When multiplying or dividing an inequality by a negative number, the inequality direction must be flipped to remain mathematically true.

Anahtar Kavram

Solving Linear Inequalities with Variables on Both Sides

Soru 4867Soru →

Match each verbal phrase with its correct algebraic expression translation.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

The phrases correctly match with their algebraic translations based on the operational keywords: 'more than' maps to addition (

y + 4

), 'less than' maps to subtraction (

y - 4

), 'product' maps to multiplication (

4y

), and 'quotient' maps to division (

\frac{y}{4}

).

Matching each phrase requires identifying the key operation word: 'more than' for addition, 'less than' for subtraction (with reversed order), 'product' for multiplication, and 'quotient' for division. Applying these directly to the variable

y

and the number 4 yields the correct algebraic expressions.

Adım Adım Çözüm

1

Translate 'Four more than a number

y

'.

y + 4

The words 'more than' specify adding to the given variable.

2

Translate 'Four less than a number

y

'.

y - 4

The words 'less than' specify subtracting from the given variable.

3

Translate 'The product of four and a number

y

'.

4y

The word 'product' specifies multiplying the values.

4

Translate 'The quotient of a number

y

and four'.

\frac{y}{4}

The word 'quotient' specifies dividing the first value by the second value.

Anahtar Kavram

Translating basic arithmetic operation keywords into their corresponding algebraic symbols.

Soru 4868Soru →

Read the passage below:

Before the advent of radar, military engineers constructed massive concrete acoustic mirrors along coastlines to detect incoming enemy aircraft by capturing engine rumble. While these monolithic structures successfully concentrated distant ambient noises, their practical utility was severely undermined by unpredictable coastal winds and background surf. Consequently, operators frequently misidentified harmless seaside commotion as approaching bombers. Though an ingenious application of basic wave mechanics, the technology ultimately proved far too finicky and erratic for dependable defense operations.

Which of the following best describes the author's tone toward early acoustic detection technology?

Unreserved praise for its technological breakthrough

Hostile outrage over the waste of military funds

Respectful recognition of its clever concept alongside clear criticism of its operational unreliability

Objective indifference toward coastal defense mechanisms

Cevabı ve açıklamayı göster

Cevap: Respectful recognition of its clever concept alongside clear criticism of its operational unreliability

Cevap

Respectful recognition of its clever concept alongside clear criticism of its operational unreliability

The correct response captures both dimensions of the author's attitude: genuine appreciation for the scientific principle ('ingenious application') combined with a frank, critical assessment of its operational flaws ('severely undermined', 'finicky and erratic').

Adım Adım Çözüm

1

Identify positive evaluative words and phrases in the passage

The author notes that the acoustic mirrors 'successfully concentrated distant ambient noises' and calls the device an 'ingenious application of basic wave mechanics'.

Tracking positive diction establishes the author's appreciation of the theoretical design.

2

Identify negative or critical evaluative language in the passage

The author describes practical utility as 'severely undermined', mentions frequent misidentifications, and labels the technology as 'far too finicky and erratic for dependable defense operations'.

Analyzing critical descriptors reveals the author's evaluation of the technology's practical shortcomings.

3

Synthesize the contrasting viewpoints to determine the overall tone

The author maintains a nuanced stance that respects the clever engineering while remaining clearly critical of its real-world performance.

Tone inference questions require balancing all descriptive elements rather than focusing on an isolated sentence.

Anahtar Kavram

Inferring Author Tone and Attitude

Tahmini Süre:1m 0s

Soru 4869Soru →

A carpenter has two pieces of wood. One piece is $18$ inches long and the other is $24$ inches long. She wants to cut both pieces into smaller segments of equal length with no wood left over. What is the greatest possible length, in inches, of each segment?

3

6

42

72

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The greatest possible length of each segment is 6 inches.

To find the greatest possible length for the cut segments, we must find the greatest common factor (GCF) of 18 and 24. The factors of 18 are 1, 2, 3, 6, 9, 18. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. The largest number appearing in both lists is 6, making it the correct answer.

Adım Adım Çözüm

1

Identify the goal of the problem.

We need to divide both 18 and 24 into equal segments of the maximum possible length. This means finding the Greatest Common Factor (GCF) of 18 and 24.

The phrase 'cut into equal length' implies finding a common factor, and 'greatest possible length' specifies the GCF.

2

List the factors of 18.

The factors of 18 are 1, 2, 3, 6, 9, and 18.

These are all the integers that divide evenly into 18.

3

List the factors of 24.

The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.

These are all the integers that divide evenly into 24.

4

Identify the greatest factor common to both lists.

The common factors are 1, 2, 3, and 6. The greatest among these is 6.

This is the largest segment size that will evenly divide both pieces of wood.

Anahtar Kavram

The Greatest Common Factor (GCF) is the largest positive integer that divides evenly into two or more given numbers. It is commonly used when splitting items into equal, maximum-sized groups.

Tahmini Süre:1m 0s

Soru 4870Soru →

During desert field maneuvers, the support company anticipated that assembling the tactical pontoon bridge would be an arduous task. Instead, because of newly issued modular locking pins, the combat engineers completed the entire structure in less than fifteen minutes without experiencing significant physical strain.

As used in the passage, what does the word "arduous" most nearly mean?

Quick and effortless

Dangerous and hostile

Demanding great physical effort and labor

Located in an arid environment

Cevabı ve açıklamayı göster

Cevap: Demanding great physical effort and labor

Cevap

Demanding great physical effort and labor

The passage uses the contrast transition word 'Instead' to set up an explicit comparison between what was expected and what actually happened. Because the bridge was assembled quickly and 'without experiencing significant physical strain,' the word 'arduous' refers to the opposite condition: something requiring strenuous effort and difficult labor.

Adım Adım Çözüm

1

Locate the target word and identify the surrounding transitional structure.

The word 'arduous' appears in the first sentence, immediately followed by the contrast transition word 'Instead' at the start of the second sentence.

Contrast transition markers indicate that the second clause presents the opposite condition of the first clause.

2

Analyze the contrasting context clue provided in the second sentence.

The actual task took 'less than fifteen minutes' and occurred 'without experiencing significant physical strain.'

Since the real outcome was quick and physically easy, the anticipated 'arduous' task must mean the direct opposite: demanding, strenuous, and requiring substantial effort.

Anahtar Kavram

Identifying Antonym and Contrast Context Clues

Tahmini Süre:1m 0s

Soru 4871Soru →

A pilot logs $120$ flight hours during the first quarter of the year. If this amount represents $40\%$ of their annual flight hour requirement, how many additional flight hours must the pilot complete by the end of the year to exactly meet the requirement?

48

72

180

300

Cevabı ve açıklamayı göster

Cevap: 180

Cevap

The pilot must complete 180 additional flight hours.

The correct answer is found by first determining the total annual requirement. Since 120 hours represents 40% of the total, the total is calculated by dividing 120 by 0.40, which equals 300 hours. The question asks for the additional hours needed, so you must subtract the 120 hours already flown from the 300 total hours, leaving exactly 180 hours.

Adım Adım Çözüm

1

Set up an equation to find the total annual flight hour requirement.

0.40 \times \text{Total} = 120

The problem states that 120 hours is exactly 40% of the full annual requirement.

2

Solve for the total annual requirement by dividing the completed hours by the decimal percentage.

\text{Total} = \frac{120}{0.40} = 300

hours

Dividing the part by the percentage (as a decimal) yields the whole base amount.

3

Subtract the hours already flown from the total annual requirement to find the remaining hours needed.

300 - 120 = 180

hours

The question specifically asks for the 'additional' flight hours required to finish the year.

Anahtar Kavram

Reverse Percentage and Successive Subtraction

Soru 4872Soru →

Evaluate the algebraic expression:

-3a^2 + 4b - |a - c|

Given the values $a = -2$ , $b = -3$ , and $c = 5$ , which of the following is the correct result?

-31

-27

-17

17

Cevabı ve açıklamayı göster

Cevap: -31

Cevap

The correct result is -31.

The expression requires precise substitution and adherence to the order of operations. Squaring the negative value

a = -2

gives

4

, which multiplied by

-3

results in

-12

. The term

4b

becomes

4(-3) = -12

. The absolute value of

a - c

evaluates to

|-2 - 5| = |-7| = 7

. Subtracting this

7

from the sum of the first two terms yields

-12 - 12 - 7 = -31

.

Adım Adım Çözüm

1

Substitute the given values for

a

,

b

, and

c

into the expression.

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

Correct substitution requires maintaining negative signs within parentheses to prevent sign errors.

2

Evaluate the exponent and the grouping symbol (absolute value).

The exponent term evaluates to

(-2)^2 = 4

. The absolute value expression evaluates to

|-7| = 7

.

According to the order of operations, exponents and operations inside grouping symbols must be resolved before multiplication.

3

Perform the multiplication.

-3(4) = -12

and

4(-3) = -12

. The expression simplifies to

-12 - 12 - 7

.

Multiplication must be executed before addition and subtraction.

4

Perform the final subtraction operations from left to right.

-12 - 12 - 7 = -24 - 7 = -31

.

Combining the remaining terms provides the final evaluated result.

Anahtar Kavram

Evaluating algebraic expressions involving negative integers, exponents, and absolute values

Tahmini Süre:1m 0s

Soru 4873Soru →

During a field exercise, a combat engineer platoon's daily fuel consumption increased by $20\%$ in the first week, and then decreased by $25\%$ in the second week. If the final fuel consumption was $4,500$ gallons per day, what was the initial daily fuel consumption in gallons?

Cevabı ve açıklamayı göster

Cevap: 5000

Cevap

5000

The correct calculation requires treating the percentage changes as sequential multipliers. The initial amount increases by

20\%

(multiplier of

1.20

) and the new amount decreases by

25\%

(multiplier of

0.75

). The combined net multiplier is

1.20 \times 0.75 = 0.90

, meaning the final value is

90\%

of the initial value. To find the initial value, we divide the final amount (

4,500

) by

0.90

, which yields

5,000

.

Adım Adım Çözüm

1

Determine the multipliers for the successive percentage changes.

A

20\%

increase is represented by

1.20

, and a

25\%

decrease is represented by

0.75

.

Successive percentage changes must be multiplied consecutively, not added together.

2

Set up an equation representing the net change.

x \times 1.20 \times 0.75 = 4500

The initial amount (

x

) multiplied by each successive change equals the final amount.

3

Calculate the combined multiplier.

1.20 \times 0.75 = 0.90

Simplifying the multipliers shows that the net change is a

10\%

decrease overall.

4

Solve the equation for the initial amount.

x = \frac{4500}{0.90} = 5000

Dividing the final consumption by the combined multiplier isolates

x

to find the original amount.

Anahtar Kavram

Successive Percentage Changes and Reverse Percentages

Soru 4874Soru →

A technician is configuring a power supply. The required output is described as twice the difference between the baseline voltage, $v$ , and six. What expression represents the required output?

2(v - 6)

2v - 6

2(6 - v)

6 - 2v

Cevabı ve açıklamayı göster

Cevap: $2(v - 6)$

Cevap

The expression

2(v - 6)

correctly represents the required output.

The phrase 'difference between the baseline voltage,

v

, and six' translates to

(v - 6)

. The word 'twice' means multiplying this entire difference by

2

. Therefore, the correct translation must use parentheses to apply the multiplication to the whole difference, resulting in

2(v - 6)

.

Adım Adım Çözüm

1

Translate 'the difference between the baseline voltage, v, and six'.

The expression becomes

v - 6

.

The word 'difference' indicates subtraction in the order the terms are given.

2

Translate 'twice' applied to the entire difference.

The expression becomes

2(v - 6)

.

Multiplying an entire binomial expression requires placing it in parentheses to ensure the multiplication applies to both terms.

Anahtar Kavram

Translating Word Phrases into Algebraic Equations

Soru 4875Soru →

A naval architect creates a scale drawing for a new submarine. The drawing uses a scale of $\frac{3}{16}\text{ inch} = 15\text{ feet}$ . If the pressure hull measures $4\frac{1}{8}\text{ inches}$ long on the drawing, what is the actual length, in feet, of the pressure hull?

Cevabı ve açıklamayı göster

Cevap: 330

Cevap

330

The scale establishes that every inch on the drawing represents

80

actual feet (

15 \div \frac{3}{16} = 80

). Multiplying the drawing length of

4\frac{1}{8}

inches (or

\frac{33}{8}

inches) by

80

feet per inch yields the correct actual length of

330

feet.

Adım Adım Çözüm

1

Find the unit rate for

1\text{ inch}

of the drawing.

15 \div \frac{3}{16} = 15 \times \frac{16}{3} = 5 \times 16 = 80\text{ feet}

per inch.

Establishing a unit rate simplifies finding actual dimensions for any measurement on the scale drawing.

2

Convert the drawing measurement from a mixed number to an improper fraction.

4\frac{1}{8} = \frac{4 \times 8 + 1}{8} = \frac{33}{8}\text{ inches}

.

Improper fractions are required to properly multiply the scale factor with the drawing measurement.

3

Calculate the actual length by multiplying the map measurement by the unit rate.

\frac{33}{8} \times 80 = 33 \times 10 = 330\text{ feet}

.

Multiplying the measurement on the drawing by the actual distance per inch yields the total actual distance.

Anahtar Kavram

Scale Drawings and Map Measurements

Soru 4876Soru →

An algebraic relationship is given by the equation:

\frac{3}{5}(10k - 25) - \frac{1}{4}(8 - 16k) = \frac{1}{2}(14k - 10) - 2(k - 9)

What value of $k$ balances the given equation?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The value that satisfies the equation is 6.

The correct value is found by carefully distributing the fractional multipliers and negative signs to eliminate all parentheses, simplifying both sides to get

10k - 17 = 5k + 13

, and finally isolating

k

by performing balanced operations to find

k = 6

.

Adım Adım Çözüm

1

Distribute the fractional coefficients into the parentheses on both sides of the equation.

(6k - 15) - (2 - 4k) = (7k - 5) - 2(k - 9)

Distribution eliminates the parentheses and clears the fractional multipliers, allowing the terms to be evaluated and combined.

2

Distribute the negative signs for the remaining terms in the equation.

6k - 15 - 2 + 4k = 7k - 5 - 2k + 18

Properly applying the negative sign ensures all terms are correctly added or subtracted, especially noting that

- \frac{1}{4}(-16k) = +4k

and

-2(-9) = +18

.

3

Combine the constant terms and the variable terms independently on each side.

10k - 17 = 5k + 13

Simplifying each side makes it easier to isolate the variable.

4

Isolate the variable term by moving all terms containing k to one side and constants to the other side.

10k - 5k = 13 + 17 \Rightarrow 5k = 30 \Rightarrow k = 6

Solving for k provides the final numerical value that balances the equation.

Anahtar Kavram

Solving complex one-variable linear equations by distributing fractional coefficients, managing negative signs, and combining like terms.

Soru 4877Soru →

A square metal plate has a side length of $3x^4$ centimeters. Which of the following expressions represents the area of the plate in square centimeters?

9x^8

6x^8

9x^6

3x^8

Cevabı ve açıklamayı göster

Cevap: $9x^8$

Cevap

9x^8

The area of a square is calculated by squaring its side length. When squaring a term like

3x^4

, the power of a product rule states that both the coefficient and the variable must be raised to the second power. Squaring the coefficient 3 yields 9, and raising

x^4

to the second power requires multiplying the exponents (

4 \times 2 = 8

), resulting in a final area of

9x^8

.

Adım Adım Çözüm

1

Recall the formula for the area of a square.

A = s^2

To find the area, the side length must be squared.

2

Substitute the given side length into the area formula.

A = (3x^4)^2

This sets up the mathematical expression that needs to be simplified.

3

Apply the power of a product rule and power of a power rule.

3^2 \cdot (x^4)^2 = 9x^8

The outer exponent (2) must be applied to both the numerical coefficient (

3^2=9

) and the variable exponent (

4 \times 2 = 8

).

Anahtar Kavram

Power of a Product and Power of a Power rules

Soru 4878Soru →

Let $x$ be a negative even integer and $y$ be a negative odd integer. Which of the following expressions will always result in a positive odd integer?

x \times y + 1

x + y

x \times y

|x| + y

Cevabı ve açıklamayı göster

Cevap: $x \times y + 1$

Cevap

x \times y + 1

When multiplying two negative integers, the result is positive. When multiplying an even integer by an odd integer, the result is always even. Therefore, the product of

x

and

y

is a positive even integer. Adding 1 to any even integer results in an odd integer. Thus, the expression always yields a positive odd integer.

Adım Adım Çözüm

1

Determine the sign and parity of the product

x \times y

.

x \times y

is a positive even integer.

Multiplying two negative numbers yields a positive number. Multiplying an even number by any integer yields an even number.

2

Evaluate the result of adding 1 to the product.

x \times y + 1

is a positive odd integer.

Adding 1 to any even integer changes its parity to odd. Adding 1 to a positive number keeps it positive.

3

Check the validity of the other provided expressions.

Only

x \times y + 1

guarantees a positive odd result.

Other expressions result in negative odd values, positive even values, or values that can fluctuate in sign based on specific magnitudes.

Anahtar Kavram

Parity and Sign Properties of Integers

Soru 4879Soru →

Read the passage below:

Invented in 1915 by a United States Forest Service employee, the Osborne Fire Finder transformed wilderness fire detection from guesswork into a rigorous science. By mounting a rotating sight ring over an exact topographic map, lookouts perched high in isolated towers could pinpoint the precise bearing and elevation of rising smoke plumes miles away. Though modern satellites and aerial drones now scan forested canopies, the rugged simplicity and remarkable precision of this mechanical tool remain a testament to pragmatic innovation under harsh field conditions.

What is the author's attitude toward the Osborne Fire Finder?

Critical of the harsh field conditions endured by isolated tower lookouts

Appreciative of its practical ingenuity and enduring effectiveness

Skeptical about its reliability compared to modern aerial drones

Indifferent, viewing it as an obsolete relic replaced by satellite technology

Cevabı ve açıklamayı göster

Cevap: Appreciative of its practical ingenuity and enduring effectiveness

Cevap

Appreciative of its practical ingenuity and enduring effectiveness

The author characterizes the device as turning detection into a 'rigorous science' and commends its 'rugged simplicity and remarkable precision' as a 'testament to pragmatic innovation.' This consistently favorable word choice demonstrates clear appreciation and respect for the invention.

Adım Adım Çözüm

1

Analyze the passage for charged descriptors and emotional diction related to the subject.

Identified positive phrases: 'rigorous science', 'rugged simplicity', 'remarkable precision', and 'testament to pragmatic innovation'.

An author's attitude is primarily revealed through qualitative adjectives and framing.

2

Evaluate the context surrounding the comparison with modern technology.

The author notes modern satellites and drones but emphasizes that the mechanical tool's precision remains impressive.

Contrasts in the text help clarify whether the author views the subject favorably or critically.

3

Select the descriptor that summarizes this stance without introducing outside assumptions.

The author holds an appreciative view rooted in respect for the tool's clever, durable design.

Mastery of tone inference requires matching the exact degree and nature of the author's sentiment.

Anahtar Kavram

Inferring Author Tone and Attitude through Textual Diction

Tahmini Süre:1m 0s

Soru 4880Soru →

Simplify the following algebraic expression completely by distributing and combining like terms:

3a(2a - 4b) - 2b(a - 5b)

Cevabı ve açıklamayı göster

Cevap: 6a^2 - 14ab + 10b^2; 6a^2-14ab+10b^2; 10b^2 - 14ab + 6a^2; 6a^2+10b^2-14ab; 6a^2 - 14ba + 10b^2

Cevap

6a^2 - 14ab + 10b^2

The expression is correctly simplified by first applying the distributive property to both parts:

3a(2a - 4b)

becomes

6a^2 - 12ab

, and

-2b(a - 5b)

becomes

-2ab + 10b^2

. Next, the like terms (

-12ab

and

-2ab

) are combined to form

-14ab

. The resulting expression in standard form is

6a^2 - 14ab + 10b^2

.

Adım Adım Çözüm

1

Distribute

3a

across the terms in the first parenthesis

(2a - 4b)

.

6a^2 - 12ab

Applying the distributive property to eliminate the first set of parentheses.

2

Distribute

-2b

across the terms in the second parenthesis

(a - 5b)

.

-2ab + 10b^2

Multiplying by a negative requires changing the signs. Specifically,

(-2b) \times (-5b) = +10b^2

.

3

Write out the full expanded expression and identify like terms.

6a^2 - 12ab - 2ab + 10b^2

Preparing to combine terms with identical variable parts.

4

Combine the like terms (

-12ab

and

-2ab

).

6a^2 - 14ab + 10b^2

-12 - 2 = -14

, resulting in the final simplified standard form.

Anahtar Kavram

Distributive Property and Combining Like Terms with Multiple Variables

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