All practice questions

Question 1701Question →

A student earned scores of $85$ , $90$ , $88$ , and $92$ on the first four quizzes of the semester. What score must the student earn on the fifth quiz to have an average (arithmetic mean) score of exactly $90$ for all five quizzes?

1

5

89

90

95

Show answer & explanation

Answer: 95

Answer

The student must earn a score of 95 on the fifth quiz.

To find the score needed on the fifth quiz to achieve an average of

90

, first find the total number of points required. An average of

90

on

5

quizzes requires a total sum of

5 \times 90 = 450

points. The sum of the first four quiz scores is

85 + 90 + 88 + 92 = 355

points. Subtracting the current sum from the required total sum (

450 - 355

) yields

95

, which is the score the student must earn on the fifth quiz.

Step-by-Step Solution

1

Calculate the sum of the student's scores on the first four quizzes.

85 + 90 + 88 + 92 = 355

To find the total number of points the student has earned so far.

2

Calculate the total sum of points required to achieve an average of

90

over five quizzes.

5 \times 90 = 450

The arithmetic mean formula is Mean = Sum / Count, which can be rearranged to Sum = Mean * Count.

3

Subtract the sum of the first four quiz scores from the required total sum of five quiz scores.

450 - 355 = 95

The difference represents the score needed on the fifth quiz to achieve the target average.

Key Concept

Finding a missing value given a target arithmetic mean

Question 1702Question →

An investment fund allocates its capital into three portfolios: Real Estate, Technology, and Green Energy. Initially, $\frac{1}{4}$ of the total capital is allocated to Real Estate, and $\frac{2}{5}$ of the total capital is allocated to Technology. The remaining capital is allocated to Green Energy. During the first year, the Real Estate portfolio decreases in value by $16\%$ , the Technology portfolio increases in value by $15\%$ , and the Green Energy portfolio increases in value by $p\%$ . If the total value of the entire investment fund at the end of the first year has increased by $9\%$ of its initial value, what is the value of $p$ ?

10.0

10.5

18.9

20.0

31.4

Show answer & explanation

Answer: $20.0$

Answer

The value of

p

is

20.0

.

The correct answer is

20.0

. By assuming a total capital of

100

units, we find that the Real Estate portfolio starts at

25

units, the Technology portfolio at

40

units, and the Green Energy portfolio at

35

units. The change in Real Estate is

-4

units, and the change in Technology is

+6

units. To reach a total fund increase of

9

units, the change in the Green Energy portfolio must satisfy

-4 + 6 + \text{Change in Green Energy} = 9

, which means the Green Energy portfolio must increase by

7

units. Finding what percent

7

is of

35

yields

\frac{7}{35} = 0.20

, or

20\%

.

Step-by-Step Solution

1

Determine the initial allocation of capital for each portfolio by assuming a total initial capital of

100

units.

Real Estate receives

\frac{1}{4} \times 100 = 25

units. Technology receives

\frac{2}{5} \times 100 = 40

units. The remaining capital allocated to Green Energy is

100 - (25 + 40) = 35

units.

This establishes the weighted share of each portfolio relative to the total capital.

2

Calculate the changes in value for the Real Estate and Technology portfolios based on their given percentage changes.

Real Estate change:

-16\% \text{ of } 25 = -0.16 \times 25 = -4

units. Technology change:

+15\% \text{ of } 40 = 0.15 \times 40 = +6

units.

This determines the absolute change in value contributed by these two portfolios.

3

Calculate the total change in value for the entire fund.

Total fund change:

+9\% \text{ of } 100 = +9

units.

The total change in the fund is the sum of the individual changes in the portfolios.

4

Set up an equation representing the sum of the changes in all three portfolios and solve for

p

.

-4 + 6 + \left(\frac{p}{100} \times 35\right) = 9 \implies 2 + 0.35p = 9 \implies 0.35p = 7 \implies p = 20

.

This solves for the unknown percentage rate of increase required for the Green Energy portfolio.

Key Concept

Weighted average of percentage changes using fractional and decimal representations.

Alternative Method

Instead of choosing an arbitrary starting capital of

100

, you can use decimals directly. Let

C

be the total capital. The weighted changes are

0.25C(-0.16) + 0.40C(0.15) + 0.35C\left(\frac{p}{100}\right) = 0.09C

. Dividing both sides by

C

yields

-0.04 + 0.06 + 0.0035p = 0.09

. Simplifying gives

0.02 + 0.0035p = 0.09 \implies 0.0035p = 0.07 \implies p = 20

.

Estimated Time:3m 0s

Question 1703Question →

At a marketing firm, the ratio of designers to writers is $3:4$ , and the ratio of writers to managers is $5:2$ . The firm hires $10$ new designers and $4$ new managers, while the number of writers remains the same. If the new ratio of designers to managers is $2:1$ , how many writers are employed at the firm?

8

20

30

40

120

Show answer & explanation

Answer: 40

Answer

There are 40 writers employed at the firm.

To find the number of writers, we combine the ratios of designers to writers (

3:4

) and writers to managers (

5:2

) by finding a common multiple for the number of writers (20). This gives a combined ratio of

15:20:8

for designers, writers, and managers. Representing these quantities as

15x

,

20x

, and

8x

, we can write the equation for the new ratio of designers to managers:

\frac{15x + 10}{8x + 4} = 2

. Solving this equation gives

15x + 10 = 16x + 8

, which simplifies to

x = 2

. The number of writers is

20(2) = 40

.

Step-by-Step Solution

1

Align the two given ratios to find a common ratio.

The ratio of designers to writers is

3:4 = 15:20

. The ratio of writers to managers is

5:2 = 20:8

. The combined ratio of designers to writers to managers is

15:20:8

.

To analyze the changes across the different categories, we need a single ratio that shares the common term (writers).

2

Express the quantities of employees in terms of a variable

x

.

Let the number of designers be

15x

, writers be

20x

, and managers be

8x

.

This allows us to represent the actual number of employees algebraically.

3

Set up an equation based on the new ratio of designers to managers after the new hires.

\frac{15x + 10}{8x + 4} = \frac{2}{1}

The firm hired 10 designers and 4 managers, and the ratio of these new quantities is

2:1

.

4

Solve the equation for

x

.

15x + 10 = 2(8x + 4) \implies 15x + 10 = 16x + 8 \implies x = 2

.

Solving for

x

gives us the multiplier to find the actual employee counts.

5

Calculate the total number of writers.

W = 20(2) = 40

.

The number of writers is represented by

20x

.

Key Concept

Combining multiple ratios and setting up algebraic equations from changing rates/proportions.

Alternative Method

Express the number of designers and managers in terms of the number of writers,

W

. The number of designers is

\frac{3}{4}W

, and the number of managers is

\frac{2}{5}W

. After hiring, the equation is

\frac{\frac{3}{4}W + 10}{\frac{2}{5}W + 4} = 2

. Solving for

W

yields

\frac{3}{4}W + 10 = \frac{4}{5}W + 8 \implies 2 = \frac{1}{20}W \implies W = 40

.

Estimated Time:2m 0s

Question 1704Question →

Four distinct quantities are defined by different exponential, radical, or scientific notation expressions. What is the correct order of these quantities from the smallest value to the largest value?

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order of the expressions from least to greatest is

5^{20}

, followed by

3^{30}

, then

8.0 \times 10^{14}

, and finally

\sqrt{2^{100}}

.

Simplifying

\sqrt{2^{100}}

yields

2^{50}

. Expressing

5^{20}

as

25^{10}

and

3^{30}

as

27^{10}

shows that

5^{20} < 3^{30}

. By estimating upper and lower bounds relative to powers of

10

, we find that

3^{30} < 30^{10} \approx 5.9 \times 10^{14}

, which is less than

8.0 \times 10^{14}

. Meanwhile,

2^{50} = 1024^5 > (10^3)^5 = 10^{15}

, which is greater than

8.0 \times 10^{14}

. This establishes the sequence

5^{20} < 3^{30} < 8.0 \times 10^{14} < \sqrt{2^{100}}

.

Step-by-Step Solution

1

Simplify the radical expression

\sqrt{2^{100}}

using fractional exponent rules.

\sqrt{2^{100}} = (2^{100})^{1/2} = 2^{50}

Converting the square root to a power of

1/2

simplifies the expression into a single base and exponent.

2

Compare the exponential expressions

5^{20}

and

3^{30}

by expressing them with a common power of

10

.

5^{20} = (5^2)^{10} = 25^{10}

and

3^{30} = (3^3)^{10} = 27^{10}

. Since

25 < 27

, then

25^{10} < 27^{10}

, so

5^{20} < 3^{30}

.

Rewriting the powers to share an exponent of

10

allows direct comparison of the base values.

3

Establish an upper limit for

3^{30}

to compare it with the scientific notation quantity

8.0 \times 10^{14}

.

3^{30} = 27^{10} < 30^{10} = 3^{10} \times 10^{10}

. Since

3^5 = 243

, we have

3^{10} = 243^2 = 59,049

. Therefore,

3^{10} \times 10^{10} = 5.9049 \times 10^{14}

. Since

5.9049 \times 10^{14} < 8.0 \times 10^{14}

, we have

3^{30} < 8.0 \times 10^{14}

.

Using a slightly larger round number base of

30

allows an upper bound calculation that demonstrates

3^{30}

is strictly less than

8.0 \times 10^{14}

.

4

Establish a lower limit for the simplified radical expression

2^{50}

to compare it with

8.0 \times 10^{14}

.

2^{50} = (2^{10})^5 = 1024^5

. Since

1024 > 1000 = 10^3

, then

1024^5 > (10^3)^5 = 10^{15}

. Because

10^{15} = 10 \times 10^{14} > 8.0 \times 10^{14}

, it follows that

2^{50} > 8.0 \times 10^{14}

.

Using the approximation

2^{10} \approx 10^3

shows that

2^{50}

is greater than

10^{15}

, which exceeds the scientific notation value.

5

Combine the individual inequalities to construct the complete chain of comparisons.

5^{20} < 3^{30} < 8.0 \times 10^{14} < \sqrt{2^{100}}

Linking the inequalities using transitive properties determines the correct least-to-greatest order.

Key Concept

Comparing complex numerical expressions containing powers, radicals, and scientific notation by finding common exponents and using base-10 estimation.

Alternative Method

Alternatively, convert all expressions into scientific notation approximations. Note that

5^{20} = 25^{10} = (2.5 \times 10)^{10} = 2.5^{10} \times 10^{10}

. Since

2.5^{10} \approx 9536

, we get

5^{20} \approx 9.5 \times 10^{13}

. Applying a similar approximation to

3^{30} = 27^{10} = 2.7^{10} \times 10^{10} \approx 2.06 \times 10^{14}

. Since

2^{50} = 1.024^5 \times 10^{15} \approx 1.13 \times 10^{15}

, we can compare the coefficients and exponents:

9.5 \times 10^{13} < 2.06 \times 10^{14} < 8.0 \times 10^{14} < 1.13 \times 10^{15}

.

Estimated Time:3m 0s

Question 1705Question →

Acoustics experts studying the ancient theater of Epidaurus have long marveled at its extraordinary sound design. Even from the back row of the massive stone structure, spectators can clearly hear a whisper from the stage. While early theories attributed this phenomenon to local wind patterns, recent research indicates that the limestone seats play a critical role, filtering out low-frequency background noise while cranking up the high-frequency voices of the performers.

Which choice best maintains the established tone of the passage?

NO CHANGE

amplifying the high-frequency voices of the performers

rendering far more effulgent the high-frequency voices of the performers

amplifying the high-frequency voices at the performers

Show answer & explanation

Answer: amplifying the high-frequency voices of the performers

Answer

amplifying the high-frequency voices of the performers

The choice that states 'amplifying the high-frequency voices of the performers' is correct because it uses the clear, objective, and academic verb 'amplifying' and the correct preposition 'of'. This language fits perfectly with the scientific, professional tone established by the surrounding sentences.

Step-by-Step Solution

1

Analyze the surrounding text to determine the established tone of the passage.

The passage uses formal, scientific, and academic vocabulary (such as 'acoustics experts,' 'marveled,' 'phenomenon,' and 'filtering out'), establishing a professional and objective tone.

Identifying the author's tone is necessary to ensure the underlined portion maintains style and tone consistency.

2

Evaluate the choices to find the one that fits this academic tone and uses correct grammar.

The phrasing 'amplifying the high-frequency voices of the performers' is clear, precise, and grammatically correct, matching the rest of the text.

This choice maintains tone consistency without being overly informal, overly pretentious, or grammatically incorrect.

Key Concept

Tone Consistency

Question 1706Question →

Unlike the visible colors of most flowers, which are produced by chemical pigments, the brilliant blue of a peacock's feather is created by structural coloration. The surface of the feather contains tiny, regular channels that reflect specific wavelengths of light. [1] Nevertheless, this microscopic architecture creates an iridescent shimmer that changes depending on the angle from which it is viewed.

Which of the following choices is the most logical transition to use at the underlined portion marked [1]?

NO CHANGE

As a result,

Similarly,

For instance,

Show answer & explanation

Answer: As a result,

Answer

As a result,

The transition 'As a result,' is correct because it establishes the logical cause-and-effect relationship between the microscopic channels on the peacock's feather and the resulting iridescent shimmer.

Step-by-Step Solution

1

Analyze the relationship between the two sentences separated by the transition.

The first sentence describes the physical structure of the feather (tiny, regular channels that reflect light), and the second sentence describes the visual outcome of this structure (an iridescent shimmer).

Understanding the logical connection between the two ideas is necessary to select the correct transitional phrase.

2

Evaluate the relationship type.

The relationship is cause-and-effect, as the physical channels directly cause the shimmer.

Identifying the relationship type allows you to eliminate transitions that denote contrast, comparison, or illustration.

3

Select the transition that expresses this cause-and-effect relationship.

'As a result,' correctly shows that the shimmer is the direct outcome of the microscopic architecture.

Matching the transition to the cause-and-effect relationship completes the logical flow of the passage.

Key Concept

Sentence-Level Transitions

Estimated Time:1m 0s

Question 1707Question →

A school district requires a chaperone-to-student ratio of $3:5$ for all field trips. If there are exactly $15$ chaperones registered for an upcoming field trip, what is the maximum number of students who can attend?

9

17

24

25

40

Show answer & explanation

Answer: $25$

Answer

The maximum number of students who can attend is

25

.

The correct answer is

25

. To maintain the required ratio of

3

chaperones to

5

students, we set up the proportion

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

, where

x

is the number of students. Cross-multiplying gives

3x = 75

, and dividing by

3

gives

x = 25

.

Step-by-Step Solution

1

Set up the proportion using the given chaperone-to-student ratio.

\frac{\text{Chaperones}}{\text{Students}} = \frac{3}{5} = \frac{15}{x}

, where

x

represents the maximum number of students.

Establishing a direct proportion allows us to find the unknown quantity while maintaining the required ratio.

2

Solve for the unknown variable

x

by cross-multiplying.

3 \times x = 5 \times 15 \implies 3x = 75

Cross-multiplication is a standard algebraic method to solve proportions.

3

Divide both sides of the equation by

3

to isolate

x

.

x = 25

This yields the final value of students who can attend.

Key Concept

Solving direct proportions from part-to-part ratios

Estimated Time:45s

Question 1708Question →

A metal alloy is composed of copper, zinc, and nickel. In the original alloy, the ratio of the weight of copper to zinc is $3:4$ , and the ratio of the weight of zinc to nickel is $4:5$ . A metallurgist melts this $240$ -gram alloy and adds a mixture of pure copper and pure nickel. The weight of the copper added is twice the weight of the nickel added. In the resulting alloy, the weight of copper is equal to the weight of nickel. What is the ratio of the weight of zinc to the total weight of the new alloy?

\frac{4}{15}

\frac{1}{3}

\frac{2}{9}

\frac{2}{7}

\frac{4}{33}

Show answer & explanation

Answer: $\frac{2}{9}$

Answer

The correct ratio of the weight of zinc to the total weight of the new alloy is

\frac{2}{9}

.

The correct answer is

\frac{2}{9}

. To find this, we first establish the joint ratio of the metals as Copper : Zinc : Nickel =

3 : 4 : 5

. For a

240

-gram alloy, this gives

60

g of copper,

80

g of zinc, and

100

g of nickel. Adding

x

grams of nickel and

2x

grams of copper yields a new copper weight of

60 + 2x

and a new nickel weight of

100 + x

. Since these weights are equal in the new alloy,

60 + 2x = 100 + x \implies x = 40

. The new total weight is

240 + 3(40) = 360

grams. The weight of zinc remains

80

grams, so the ratio of zinc to the total weight is

80/360 = \frac{2}{9}

.

Step-by-Step Solution

1

Express the ratio of the three metals in the original alloy.

Copper : Zinc : Nickel =

3 : 4 : 5

Since Copper : Zinc =

3:4

and Zinc : Nickel =

4:5

, the zinc terms are already aligned, allowing us to combine them into a single three-part ratio.

2

Calculate the initial weights of copper, zinc, and nickel in the

240

-gram alloy.

Copper =

60

grams, Zinc =

80

grams, Nickel =

100

grams

The sum of the ratio parts is

3 + 4 + 5 = 12

. Each part represents

240 \text{ grams} / 12 = 20

grams. Multiplying each part by 20 gives the initial weights.

3

Define variables for the added metals and set up the equation for the new weights.

New Copper =

60 + 2x

, New Nickel =

100 + x

Let

x

represent the weight of nickel added in grams. Since the weight of the copper added is twice that of the nickel added,

2x

grams of copper are added.

4

Solve for the value of

x

using the condition that the final weights of copper and nickel are equal.

x = 40

Setting the expressions equal gives

60 + 2x = 100 + x

. Subtracting

x

and

60

from both sides yields

x = 40

grams.

5

Determine the new total weight of the alloy and calculate the final ratio of zinc to the total weight.

New Total Weight =

360

grams, Final Ratio =

\frac{2}{9}

The new total weight is the original weight plus the added metals:

240 + 3(40) = 360

grams. Since no zinc was added, the weight of zinc remains

80

grams. The ratio is

80 / 360 = \frac{2}{9}

.

Key Concept

Combining ratios and setting up algebraic equations to model changing quantities in mixtures.

Question 1709Question →

The pistol shrimp relies on a specialized claw to hunt. When snapped shut, the claw produces a high-pressure bubble of water that collapses with a loud snap, producing a flash of light that is visible to the eye and can be seen.

Which choice most effectively eliminates redundancy in the underlined portion of the sentence?

NO CHANGE

producing a flash of light

it produces a flash of light

producing a flash of light. Which is visible

Show answer & explanation

Answer: producing a flash of light

Answer

producing a flash of light

The correct option is the most concise choice because it removes the redundant phrases 'visible to the eye' and 'can be seen,' which repeat the same idea of visibility. It leaves a grammatically correct and direct description of the flash of light.

Step-by-Step Solution

1

Identify any redundant or repetitive phrasing in the underlined portion of the sentence.

The phrase 'visible to the eye' and the clause 'can be seen' both describe the ability of the flash to be observed, making them redundant when used together.

ACT English questions require eliminating wordiness and synonym redundancy to make sentences as clear and concise as possible.

2

Evaluate the options to find the one that is most concise while maintaining proper sentence structure.

The option 'producing a flash of light' eliminates the redundant descriptions entirely. The options containing 'it produces' and '. Which is visible' introduce grammatical errors (a comma splice and a sentence fragment, respectively).

The correct answer must be both grammatically correct and free of unnecessary repetition.

Key Concept

Eliminating Redundancy

Question 1710Question →

If $a = 2 \times 10^3$ and $b = 3 \times 10^{-4}$ , what is the value of the expression $\sqrt{\frac{a^3 \cdot b^2}{1.8}}$ ?

Show answer & explanation

Answer: 20

Answer

20

Evaluating the components of the expression sequentially:

a^3 = 8 \times 10^9

and

b^2 = 9 \times 10^{-8}

. Multiplying them yields

(8 \times 9) \times 10^{9-8} = 72 \times 10^1 = 720

. Dividing by the decimal

1.8

results in

720 / 1.8 = 400

. The square root of

400

yields

20

.

Step-by-Step Solution

1

Calculate the value of

a^3

8 \times 10^9

Apply the power of a product rule,

(xy)^n = x^n y^n

, and the power of a power rule,

(10^p)^q = 10^{p \cdot q}

.

2

Calculate the value of

b^2

9 \times 10^{-8}

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

(3 \times 10^{-4})^2

.

3

Multiply

a^3

and

b^2

720

Multiply the coefficients (

8 \times 9 = 72

) and add the exponents of the base 10 (

10^9 \times 10^{-8} = 10^1

), resulting in

72 \times 10 = 720

.

4

Divide the product by

1.8

400

Substitute the values to get

720 / 1.8

, which simplifies to

7200 / 18 = 400

.

5

Take the square root of the quotient

20

Find the non-negative square root of

400

, which is

20

.

Key Concept

Simplifying numerical expressions containing combinations of exponent rules, roots, and scientific notation.

Estimated Time:1m 30s

Question 1711Question →

The following sentences are from a paragraph about nurse logs in temperate rainforests. In what order should these sentences be arranged to create the most logical and cohesive paragraph?

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct sequence of sentences is: first, the sentence introducing the death of a massive tree as a new cycle of life; second, the sentence identifying these trees as nurse logs; third, the sentence explaining that seedlings sprout on the decaying wood; and fourth, the concluding sentence summarizing that debris becomes the foundation of the forest.

The correct sequence begins with the broad introductory statement about a tree's death initiating new life, transitions to naming these trees 'nurse logs' and describing their nutrients, details the seedlings sprouting as a direct result of these nutrients, and concludes with a summary sentence showing how debris becomes the foundation of the future forest.

Step-by-Step Solution

1

Identify the introductory sentence.

The sentence introducing the death of a massive tree as the beginning of a new cycle of life establishes the paragraph's topic without referencing any outside context.

A logical paragraph must begin with a clear topic sentence that introduces the main subject.

2

Link the second sentence to the introduction using transition clues.

The sentence beginning with 'These fallen giants' directly refers back to the 'death of a massive tree' and names them 'nurse logs'.

Cohesion requires that demonstrative pronouns like 'these' refer back to an established antecedent in the preceding sentence.

3

Connect the supporting detail sentence.

The sentence starting with 'As a result' describes how seedlings sprout on the decaying wood, which explains the specific benefit of the nutrients and moisture mentioned in the previous sentence.

Cause-and-effect transitions must logically connect the cause (nutrients and moisture) with the effect (seedlings sprouting).

4

Place the concluding sentence.

The sentence starting with 'Consequently' summarizes the paragraph's transition from forest debris to the foundation of the future forest.

Concluding sentences often begin with summary transitions like 'consequently' to wrap up the main idea.

Key Concept

Opening and Concluding Sentences

Estimated Time:1m 30s

Question 1712Question →

A scientist estimates that the population of a certain bacteria culture doubles every 4 hours. If the culture starts with $3.0 \times 10^5$ bacteria, the population after 24 hours can be written in scientific notation as $a \times 10^7$ . What is the value of $a$ ?

Show answer & explanation

Answer: 1.92

Answer

The value of the coefficient

a

is

1.92

.

The correct answer is

1.92

. In 24 hours, a population that doubles every 4 hours undergoes 6 doubling cycles. The growth factor is

2^6 = 64

. Multiplying the initial population of

3.0 \times 10^5

by 64 yields

192 \times 10^5

. Converting this value into standard scientific notation gives

1.92 \times 10^7

. Comparing this expression to the format

a \times 10^7

shows that the coefficient

a

equals

1.92

.

Step-by-Step Solution

1

Determine the number of doubling periods.

6 doubling periods

Since the population doubles every 4 hours, over a span of 24 hours it will double

24 \div 4 = 6

times.

2

Calculate the growth multiplier.

64

Doubling 6 times is represented by the exponential expression

2^6

, which evaluates to 64.

3

Multiply the initial population by the growth factor.

192 \times 10^5

Multiply the coefficient of the initial population by the growth multiplier:

3.0 \times 64 = 192

, yielding a total population of

192 \times 10^5

.

4

Convert the resulting value to the required scientific notation format.

1.92 \times 10^7

To express

192 \times 10^5

in the standard scientific notation format

a \times 10^7

, divide 192 by 100 to get 1.92, and multiply the power of 10 by

10^2

(raising the exponent from 5 to 7).

Key Concept

Evaluating exponential growth and converting numbers to standard scientific notation format

Estimated Time:1m 30s

Question 1713Question →

Which of the following is the completely factored form of $x^2 - 64$ ?

(x - 8)^2

(x - 32)(x + 32)

(x - 8)(x + 8)

(x^2 - 8)(x^2 + 8)

(x - 16)(x + 4)

Show answer & explanation

Answer: $(x - 8)(x + 8)$

Answer

(x - 8)(x + 8)

The polynomial

x^2 - 64

is a difference of two squares because

x^2

is the square of

x

and

64

is the square of

8

. Applying the difference of squares formula,

a^2 - b^2 = (a - b)(a + b)

, with

a = x

and

b = 8

gives the factored form

(x - 8)(x + 8)

.

Step-by-Step Solution

1

Identify the structure of the polynomial.

The expression

x^2 - 64

is a difference of two squares since

x^2

is

(x)^2

and

64

is

(8)^2

.

Recognizing the difference of squares allows the use of the algebraic identity

a^2 - b^2 = (a - b)(a + b)

.

2

Apply the difference of squares identity with

a = x

and

b = 8

.

(x - 8)(x + 8)

Substituting

x

and

8

into the formula yields the factored form.

Key Concept

Factoring the difference of squares using the identity

a^2 - b^2 = (a - b)(a + b)

Question 1714Question →

Which of the following is equivalent to the expression $4x^2(x - 2y) - 3x(x^2 - 3xy) - (x^3 + xy^2)$ for all real values of $x$ and $y$ ?

x^2y + xy^2

-17x^2y - xy^2

-x^3 + x^2 + x^2y - xy^2

x^2y - xy^2

-17x^2y + xy^2

Show answer & explanation

Answer: $x^2y - xy^2$

Answer

The correct simplified expression is

x^2y - xy^2

.

Distributing the terms yields

4x^3 - 8x^2y - 3x^3 + 9x^2y - x^3 - xy^2

. Grouping and combining like terms results in

(4-3-1)x^3 + (-8+9)x^2y - xy^2 = x^2y - xy^2

.

Step-by-Step Solution

1

Distribute the term

4x^2

to the terms inside the first set of parentheses:

4x^2(x - 2y)

.

4x^3 - 8x^2y

Multiplying a monomial by a binomial requires distributing the multiplier to each term inside.

2

Distribute the term

-3x

to the terms inside the second set of parentheses:

-3x(x^2 - 3xy)

.

-3x^3 + 9x^2y

Carefully apply the distributive property, remembering that multiplying a negative term by a negative term yields a positive term:

-3x \cdot (-3xy) = 9x^2y

.

3

Distribute the negative sign (or

-1

) to the terms inside the third set of parentheses:

-(x^3 + xy^2)

.

-x^3 - xy^2

Distributing the negative sign changes the signs of all terms inside the parentheses.

4

Combine all parts and group the like terms:

x^3

,

x^2y

, and

xy^2

.

x^2y - xy^2

Add the coefficients of the terms with the same variable parts:

(4 - 3 - 1)x^3 + (-8 + 9)x^2y - xy^2 = 0x^3 + x^2y - xy^2 = x^2y - xy^2

.

Key Concept

Simplifying algebraic expressions by distributing and combining like terms.

Estimated Time:1m 0s

Question 1715Question →

For the imaginary unit $i$ , which of the following is equivalent to the complex expression $\frac{2 + i^{15}}{1 - 2i^9}$ ?

\frac{4}{5} + \frac{3}{5}i

i

\frac{3}{5}i

\frac{4}{5}

-\frac{4}{3} - i

Show answer & explanation

Answer: $\frac{4}{5} + \frac{3}{5}i$

Answer

\frac{4}{5} + \frac{3}{5}i

Simplifying the powers of

i

yields

i^{15} = -i

and

i^9 = i

, giving the expression

\frac{2 - i}{1 - 2i}

. Multiplying the numerator and denominator by the complex conjugate

1 + 2i

results in the fraction

\frac{(2 - i)(1 + 2i)}{(1 - 2i)(1 + 2i)}

. Expanding both parts and substituting

i^2 = -1

gives

\frac{4 + 3i}{5}

, which simplifies to

\frac{4}{5} + \frac{3}{5}i

.

Step-by-Step Solution

1

Simplify the powers of the imaginary unit

i

in the expression.

i^{15} = i^{12} \cdot i^3 = 1 \cdot (-i) = -i

and

i^9 = i^8 \cdot i = 1 \cdot i = i

. The expression becomes

\frac{2 - i}{1 - 2i}

.

Reducing powers of

i

simplifies the expression and makes it easier to work with binomials.

2

Multiply the numerator and denominator by the complex conjugate of the denominator,

1 + 2i

.

\frac{2 - i}{1 - 2i} \cdot \frac{1 + 2i}{1 + 2i} = \frac{(2 - i)(1 + 2i)}{(1 - 2i)(1 + 2i)}

Multiplying by the conjugate rationalizes the denominator, converting it to a real number.

3

Expand and simplify the numerator and denominator using the property

i^2 = -1

.

Numerator:

(2 - i)(1 + 2i) = 2 + 4i - i - 2i^2 = 2 + 3i - 2(-1) = 4 + 3i

. Denominator:

(1 - 2i)(1 + 2i) = 1 - 4i^2 = 1 - 4(-1) = 5

.

Expanding the binomial products allows combining real and imaginary parts.

4

Write the resulting fraction in standard complex form

a + bi

.

\frac{4 + 3i}{5} = \frac{4}{5} + \frac{3}{5}i

Standard form separates the real part and the imaginary part clearly.

Key Concept

Simplifying complex expressions by reducing powers of

i

and rationalizing the denominator using the complex conjugate.

Question 1716Question →

What is the positive difference between the two real solutions to the quadratic equation $2x(3x - 5) = 3(x - 2)$ ?

\frac{1}{6}

1

\frac{5}{6}

\frac{11}{6}

\frac{13}{6}

Show answer & explanation

Answer: $\frac{5}{6}$

Answer

The positive difference between the two solutions is

\frac{5}{6}

.

The correct positive difference is

\frac{5}{6}

because rearranging the equation

2x(3x - 5) = 3(x - 2)

yields

6x^2 - 13x + 6 = 0

. Factoring this equation gives

(2x - 3)(3x - 2) = 0

, which has the solutions

x = \frac{3}{2}

and

x = \frac{2}{3}

. Subtracting these values gives

\frac{3}{2} - \frac{2}{3} = \frac{9 - 4}{6} = \frac{5}{6}

.

Step-by-Step Solution

1

Expand both sides of the equation.

6x^2 - 10x = 3x - 6

To begin solving, distribute the term

2x

on the left side and the constant

3

on the right side.

2

Rearrange the terms to set the quadratic equation to zero.

6x^2 - 13x + 6 = 0

Subtract

3x

and add

6

to both sides of the equation to write it in standard form

ax^2 + bx + c = 0

.

3

Factor the quadratic equation over the integers.

(2x - 3)(3x - 2) = 0

Find two binomials whose product is

6x^2 - 13x + 6

. We search for two numbers that multiply to

36

(from

6 \times 6

) and sum to

-13

, which are

-9

and

-4

, allowing factoring by grouping.

4

Solve for the roots of the equation.

x = \frac{3}{2}

or

x = \frac{2}{3}

Set each linear factor equal to zero using the Zero Product Property:

2x - 3 = 0 \implies x = \frac{3}{2}

and

3x - 2 = 0 \implies x = \frac{2}{3}

.

5

Calculate the positive difference between the two solutions.

\frac{3}{2} - \frac{2}{3} = \frac{9}{6} - \frac{4}{6} = \frac{5}{6}

Subtract the smaller solution from the larger solution to find the positive difference.

Key Concept

Solving quadratic equations by rearranging and factoring over integers.

Question 1717Question →

For all real numbers $x$ and $y$ , which of the following is equivalent to the expression $(x - 2y)^2 - (x^2 - 5xy)$ ?

4y + xy

4y^2 - 9xy

4y^2 + 5xy

4y^2 + xy

4y^2 - 5xy

Show answer & explanation

Answer: $4y^2 + xy$

Answer

The expression is equivalent to

4y^2 + xy

.

Expanding

(x - 2y)^2

yields

x^2 - 4xy + 4y^2

. Distributing the negative sign to

-(x^2 - 5xy)

gives

-x^2 + 5xy

. Combining the results yields

(x^2 - x^2) + (-4xy + 5xy) + 4y^2 = xy + 4y^2

, which is equivalent to

4y^2 + xy

.

Step-by-Step Solution

1

Expand the squared binomial

(x - 2y)^2

.

x^2 - 4xy + 4y^2

Apply the binomial squaring formula

(a - b)^2 = a^2 - 2ab + b^2

to expand the expression.

2

Distribute the negative sign across the second parenthetical expression

-(x^2 - 5xy)

.

-x^2 + 5xy

Distributing the negative sign changes the signs of both terms inside the parentheses.

3

Combine like terms from the expanded parts.

xy + 4y^2

Group and add coefficients of the like terms:

(x^2 - x^2) + (-4xy + 5xy) + 4y^2 = 0 + xy + 4y^2

.

Key Concept

Simplifying Expressions and Combining Like Terms

Question 1718Question →

Which of the following expressions represents the complete factorization of $2x^4 - 32$ ?

2(x^2 - 4)(x^2 + 4)

(2x^2 - 8)(x^2 + 4)

2(x - 2)(x + 2)(x + 2)^2

2(x - 2)(x + 2)(x^2 + 4)

2(x - 4)(x + 4)

Show answer & explanation

Answer: $2(x - 2)(x + 2)(x^2 + 4)$

Answer

2(x - 2)(x + 2)(x^2 + 4)

The correct answer is found by first factoring out the greatest common factor of

2

, resulting in

2(x^4 - 16)

. Next, recognize that

x^4 - 16

is a difference of squares,

(x^2)^2 - 4^2

, which factors into

(x^2 - 4)(x^2 + 4)

. Finally, factor the remaining difference of squares,

x^2 - 4

, into

(x - 2)(x + 2)

. The sum of squares,

x^2 + 4

, cannot be factored further. Combining these parts gives the complete factorization.

Step-by-Step Solution

1

Factor out the greatest common factor from the polynomial.

2(x^4 - 16)

Both terms of

2x^4 - 32

are divisible by

2

, so we factor it out to simplify the remaining expression.

2

Factor the difference of squares inside the parentheses.

2(x^2 - 4)(x^2 + 4)

The expression

x^4 - 16

is a difference of squares because it can be written as

(x^2)^2 - 4^2

.

3

Factor the remaining difference of squares.

2(x - 2)(x + 2)(x^2 + 4)

The binomial

x^2 - 4

is also a difference of squares (

x^2 - 2^2

), which factors into

(x - 2)(x + 2)

. The sum of squares

x^2 + 4

cannot be factored further using real numbers.

Key Concept

Factoring polynomials completely by extracting the greatest common factor and repeatedly applying the difference of squares formula.

Estimated Time:1m 0s

Question 1719Question →

The square of $3$ less than a number $x$ is equal to $16$ . What is the greater of the two possible values for $x$ ?

5

11

7

19

25

Show answer & explanation

Answer: 7

Answer

7

The word problem translates to the equation

(x-3)^2 = 16

. Expanding the left side and subtracting 16 from both sides gives the standard quadratic equation

x^2 - 6x - 7 = 0

. This factors into

(x-7)(x+1) = 0

, yielding two solutions:

x = 7

and

x = -1

. The greater of these two values is 7.

Step-by-Step Solution

1

Translate the verbal description into an algebraic equation.

(x-3)^2 = 16

'3 less than a number

x

' is written as

x - 3

, and its square is set equal to 16.

2

Expand the squared binomial and rearrange the terms into standard quadratic form

ax^2 + bx + c = 0

.

x^2 - 6x - 7 = 0

Expanding

(x-3)^2

yields

x^2 - 6x + 9

. Subtracting 16 from both sides sets the equation to zero.

3

Factor the quadratic equation over the integers.

(x - 7)(x + 1) = 0

We find two integers that multiply to

-7

and add to

-6

, which are

-7

and

1

.

4

Solve for

x

by setting each factor to zero, then select the greater value.

x = 7

(since the solutions are

7

and

-1

)

Setting the factors to zero gives

x - 7 = 0 \implies x = 7

and

x + 1 = 0 \implies x = -1

. The larger of these two values is

7

.

Key Concept

Solving Quadratic Equations by Factoring

Estimated Time:1m 0s

Question 1720Question →

Which of the following is a factor of the polynomial $8x^3 - 12x^2 - 18x + 27$ ?

2x - 9

4x - 9

4x - 3

2x + 3

2x + 9

Show answer & explanation

Answer: $2x + 3$

Answer

The correct answer is

2x + 3

.

The correct answer is the factor

2x + 3

. By grouping the first two terms and the last two terms, we get

4x^2(2x - 3) - 9(2x - 3)

. Factoring out the common binomial yields

(4x^2 - 9)(2x - 3)

. The term

4x^2 - 9

is a difference of squares, which factors into

(2x - 3)(2x + 3)

. Therefore, the fully factored expression is

(2x - 3)^2(2x + 3)

, making the linear expression with a positive constant term the correct factor.

Step-by-Step Solution

1

Group the four terms of the polynomial into two pairs.

(8x^3 - 12x^2) - (18x - 27)

Grouping terms allows us to find common binomial factors in a cubic polynomial.

2

Factor out the greatest common factor (GCF) from each group.

4x^2(2x - 3) - 9(2x - 3)

The GCF of

8x^3

and

12x^2

is

4x^2

, and the GCF of

-18x

and

27

is

-9

(factoring out the negative to match the binomials).

3

Factor out the common binomial factor

(2x - 3)

.

(4x^2 - 9)(2x - 3)

Both groups share the factor

(2x - 3)

, so it can be factored out.

4

Factor the quadratic difference of squares

(4x^2 - 9)

.

(2x - 3)(2x + 3)(2x - 3) = (2x - 3)^2(2x + 3)

The term

4x^2 - 9

is a difference of squares of the form

a^2 - b^2 = (a-b)(a+b)

, where

a = 2x

and

b = 3

.

Key Concept

Factoring a cubic polynomial completely by grouping and then applying the difference of squares formula.

Answer

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