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Question 1581Question →

If x=−6x = -6, what is the value of the expression 12−∣x−3∣12 - |x - 3|?

Show answer & explanation

Answer: 3

Answer

The value of the expression is 3.
Substituting x=−6x = -6 into the expression 12−∣x−3∣12 - |x - 3| gives 12−∣−6−3∣12 - |-6 - 3|. Simplifying the subtraction inside the absolute value gives 12−∣−9∣12 - |-9|. Since the absolute value of −9-9 is 99, the expression simplifies to 12−912 - 9, which equals 33.

Step-by-Step Solution

1
Substitute x=−6x = -6 into the expression.
12−∣−6−3∣12 - |-6 - 3|
Substitute the given value of the variable into the algebraic expression.
2
Simplify the expression inside the absolute value symbol.
12−∣−9∣12 - |-9|
Subtracting 3 from −6-6 results in −9-9.
3
Calculate the absolute value.
12−912 - 9
The absolute value represents the distance from zero on the number line, so ∣−9∣=9|-9| = 9.
4
Subtract the numbers to find the final value.
33
12−9=312 - 9 = 3.

Key Concept

Evaluating algebraic expressions involving integers and absolute values.
Question 1582Question →

The passage below is adapted from an article about document conservation.

When archivist Amanda Stone first examined the nineteenth-century ship logs, she noticed that the ink had begun to fade under the relentless humidity. To prevent further decay, she recommended that the museum store the documents in a climate-controlled vault. By maintaining a constant temperature and minimal moisture, the staff could arrest the degradation of the delicate fibers. This preservation effort ensured that future generations of historians could study these primary sources.

Which choice most precisely and clearly fits the context of the sentence?

Show answer & explanation

Answer: NO CHANGE

Answer

The original phrasing 'NO CHANGE' (using the word 'arrest') is the correct choice because it precisely describes stopping or checking the progress of the degradation of the delicate fibers.
The word 'arrest' is correct because, in a formal or scientific context, it precisely means to stop or check the progress of a process, such as the degradation of delicate fibers.

Step-by-Step Solution

1
Analyze the sentence context to determine the intended meaning of the underlined word.
The sentence explains how maintaining a constant temperature and minimal moisture can stop or check the progress of the degradation of the delicate fibers.
Understanding the context helps identify the precise action needed, which is to halt a physical or chemical process.
2
Evaluate the options to identify the word that most precisely fits the context of stopping a physical process.
The word 'arrest' is a precise term used to describe stopping or checking the progress of a process. The other options are synonyms in the context of law enforcement or physical seizing but do not fit the scientific/conservation register.
Determining which word has the correct connotation and register ensures word precision and clarity.

Key Concept

Word Precision and Clarity
Question 1583Question →

In 1913, journalist Arthur Wynne created a new game for the *New York World* newspaper. Initially called a 'Word-Cross,' this puzzle was shaped like a diamond with a hollow center. Wynne's creation proved to be incredibly popular, and it established a precedent that would soon go on to become a staple of newspapers across the country.

Which of the following is the best revision of the underlined portion?

Show answer & explanation

Answer: soon became a staple of newspapers across the country.

Answer

soon became a staple of newspapers across the country.
The correct answer is the most concise option that preserves the original meaning and is grammatically correct. Replacing the wordy clause with the active verb phrase 'soon became' makes the sentence direct and clear.

Step-by-Step Solution

1
Identify the underlined portion and evaluate its conciseness and clarity.
The phrase 'established a precedent that would soon go on to become' is overly wordy and contains redundant ideas.
ACT English questions prioritize clear, concise writing that avoids unnecessary words.
2
Evaluate the options for both grammatical correctness and conciseness.
The option that says 'soon became a staple of newspapers across the country' is active and direct. The other options either retain the wordiness or introduce grammatical errors, such as creating a sentence fragment or adding incorrect commas.
Selecting the most concise, grammatically correct option ensures the sentence is clear and effective.

Key Concept

Eliminating Wordiness
Estimated Time:45s
Question 1584Question →

A positive integer nn has the prime factorization 2a×3b×5c2^a \times 3^b \times 5^c, where aa, bb, and cc are positive integers. The greatest common divisor of nn and 840840 is 120120, and the least common multiple of nn and 9090 is 1,8001,800. If nn is not divisible by 99, what is the value of a+b+ca + b + c?

Show answer & explanation

Answer: 6

Answer

6
The correct answer is 6. By analyzing the prime factorizations, the greatest common divisor condition GCD(n,840)=120\text{GCD}(n, 840) = 120 tells us that the exponents of the prime factors of nn satisfy a≥3a \geq 3, b≥1b \geq 1, and c≥1c \geq 1. The least common multiple condition LCM(n,90)=1,800\text{LCM}(n, 90) = 1,800 tells us that a=3a = 3, b≤2b \leq 2, and c=2c = 2. Finally, the condition that nn is not divisible by 9 requires b<2b < 2, which forces b=1b = 1. Summing these values gives 3+1+2=63 + 1 + 2 = 6.

Step-by-Step Solution

1
Find the prime factorizations of the given numbers.
840=23×31×51×71840 = 2^3 \times 3^1 \times 5^1 \times 7^1, 120=23×31×51120 = 2^3 \times 3^1 \times 5^1, 90=21×32×5190 = 2^1 \times 3^2 \times 5^1, and 1,800=23×32×521,800 = 2^3 \times 3^2 \times 5^2.
Decomposing the numbers into their prime factors allows us to apply the rules of Greatest Common Divisor (GCD) and Least Common Multiple (LCM) on the exponents.
2
Analyze the GCD condition.
Since GCD(n,840)=120\text{GCD}(n, 840) = 120, we have min⁡(a,3)=3\min(a, 3) = 3, min⁡(b,1)=1\min(b, 1) = 1, and min⁡(c,1)=1\min(c, 1) = 1. This implies a≥3a \geq 3, b≥1b \geq 1, and c≥1c \geq 1.
The GCD of two numbers takes the minimum exponent for each prime factor present in both numbers.
3
Analyze the LCM condition.
Since LCM(n,90)=1,800\text{LCM}(n, 90) = 1,800, we have max⁡(a,1)=3\max(a, 1) = 3, max⁡(b,2)=2\max(b, 2) = 2, and max⁡(c,1)=2\max(c, 1) = 2. This implies a=3a = 3, b≤2b \leq 2, and c=2c = 2.
The LCM of two numbers takes the maximum exponent for each prime factor present in either number.
4
Apply the divisibility constraint to determine the exponents.
From previous steps, a=3a = 3 and c=2c = 2. The exponent bb must satisfy 1≤b≤21 \leq b \leq 2. Since nn is not divisible by 9=329 = 3^2, the exponent of 3 in the factorization of nn must be strictly less than 2. Thus, b=1b = 1.
If b=2b = 2, then nn would contain 323^2, making it divisible by 9, which violates the given constraint.
5
Calculate the sum a+b+ca + b + c.
a+b+c=3+1+2=6a + b + c = 3 + 1 + 2 = 6.
Adding the determined exponents yields the final requested value.

Key Concept

Using prime factorizations to find the greatest common divisor and least common multiple of integers.

Alternative Method

Instead of checking each prime factor independently, we can find the value of nn directly. Since GCD(n,840)=120\text{GCD}(n, 840) = 120, nn must be a multiple of 120120. The multiples of 120120 are 120,240,360,480,600,720,840,…120, 240, 360, 480, 600, 720, 840, \dots. We test which of these multiples satisfies LCM(n,90)=1,800\text{LCM}(n, 90) = 1,800 and is not divisible by 99. Since 1,800=15×1201,800 = 15 \times 120, nn must divide 1,8001,800. The divisors of 1,8001,800 that are multiples of 120120 are 120,240,360,600,900,1,800120, 240, 360, 600, 900, 1,800. Among these, only 600600 has a greatest common divisor of 120120 with 840840, is not divisible by 99, and has LCM(600,90)=1,800\text{LCM}(600, 90) = 1,800. Thus, n=600=23×31×52n = 600 = 2^3 \times 3^1 \times 5^2, which gives a+b+c=3+1+2=6a+b+c = 3+1+2 = 6.
Estimated Time:2m 0s
Question 1585Question →

Biologists studying New Caledonian crows have observed that these birds can fashion hooks from twigs. By manufacturing and creating these tools, the crows can easily extract insects from deep within tree bark.

Which of the following options represents the most effective way to write the underlined portion of the sentence?

Show answer & explanation

Answer: By manufacturing these tools,

Answer

The correct option is the phrase that removes the redundant word, resulting in 'By manufacturing these tools,'.
The correct option removes the word 'creating' from the introductory phrase, which is redundant because it means the same thing as 'manufacturing' in this context. The resulting phrase is concise, maintains the intended meaning, and links correctly to the main clause of the sentence.

Step-by-Step Solution

1
Identify the redundancy in the original underlined phrase.
The phrase contains both 'manufacturing' and 'creating', which are synonymous in this context and repeat the same meaning unnecessarily.
ACT English questions require eliminating wordiness and redundancy to make sentences as concise as possible.
2
Evaluate the remaining options for conciseness and grammatical correctness.
The option containing the semicolon and relative pronoun creates a fragment. The option with 'They manufacture these tools' creates a comma splice. The option with just 'By manufacturing these tools,' is both concise and grammatically correct.
Eliminating grammatically incorrect distractors ensures the final choice is correct in both punctuation and syntax.

Key Concept

Eliminating Redundancy
Question 1586Question →

A certain type of single-celled organism has a length of approximately 2.5×10−62.5 \times 10^{-6} meters. If 4×1034 \times 10^3 of these organisms are placed end-to-end in a straight line, what is the total length of the line, in meters, written in scientific notation?

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Answer: 1.0×10−21.0 \times 10^{-2}

Answer

The total length of the line is 1.0×10−21.0 \times 10^{-2} meters.
To find the total length of the line of organisms, multiply the length of a single organism by the total number of organisms: (2.5×10−6)×(4×103)(2.5 \times 10^{-6}) \times (4 \times 10^3). Grouping the coefficients and powers of ten gives (2.5×4)×(10−6×103)=10×10−3(2.5 \times 4) \times (10^{-6} \times 10^3) = 10 \times 10^{-3}. To write this in standard scientific notation, rewrite 1010 as 1.0×1011.0 \times 10^1, which yields 1.0×101+(−3)=1.0×10−21.0 \times 10^{1 + (-3)} = 1.0 \times 10^{-2}.

Step-by-Step Solution

1
Set up the multiplication of the two values to find the total length.
Total Length =(2.5×10−6)×(4×103)= (2.5 \times 10^{-6}) \times (4 \times 10^3)
To find the combined length of multiple organisms placed end-to-end, multiply the length of one organism by the total number of organisms.
2
Group the coefficients and the powers of 1010 together, then perform the multiplication.
(2.5×4)×(10−6×103)=10×10−6+3=10×10−3(2.5 \times 4) \times (10^{-6} \times 10^3) = 10 \times 10^{-6 + 3} = 10 \times 10^{-3}
Use the associative and commutative properties of multiplication, and add the exponents when multiplying powers with the same base: 10a×10b=10a+b10^a \times 10^b = 10^{a+b}.
3
Convert the result into standard scientific notation a×10na \times 10^n where 1≤∣a∣<101 \le |a| < 10.
10×10−3=1.0×101×10−3=1.0×10−210 \times 10^{-3} = 1.0 \times 10^1 \times 10^{-3} = 1.0 \times 10^{-2}
Since 1010 is not less than 1010, rewrite 1010 as 1.0×1011.0 \times 10^1 and add the exponents (1+(−3)=−21 + (-3) = -2) to express the number in standard scientific notation.

Key Concept

Multiplying numbers in scientific notation and converting to standard scientific notation.
Estimated Time:1m 0s
Question 1587Question →

Read the passage below about elephant migration. Which precise and clear word should be placed in the blank to describe the matriarch's comprehensive, far-reaching knowledge of the terrain?

Fill in the blanks below

During the dry season, African elephants migrate long distances to find water. These intelligent animals rely on their memories to locate watering holes they visited years before. The matriarch leads the herd, using her knowledge of the terrain to guide them safely.
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Answer

extensive (or a synonym like vast, immense, or broad)
The correct fill-in word must precisely and clearly describe the matriarch's far-reaching and comprehensive knowledge of the terrain. The word 'extensive' fits the academic register and context perfectly, as do synonyms like 'vast', 'immense', or 'broad'.

Step-by-Step Solution

1
Analyze the context clues in the passage.
The sentence notes that the elephants migrate long distances and rely on memories of watering holes from years ago.
This establishes that the matriarch's knowledge covers large geographic areas and long spans of time.
2
Evaluate word choices for precision, clarity, and register.
A word like 'extensive' or 'vast' accurately describes this comprehensive scope of knowledge without violating the formal tone of the passage.
Vague or informal words fail to convey the scale of the elephant's cognitive map.

Key Concept

Word Precision and Clarity
Question 1588Question →

A writer is editing a draft of an essay about photographic history.

[1] In 1842, astronomer and chemist Sir John Herschel invented the cyanotype, a photographic printing process that produces a Prussian blue monochrome image. [2] Herschel was primarily looking for a reliable way to copy his extensive mathematical and astronomical notes without having to rewrite them by hand. [3] The process involves coating paper with a solution of ammonium iron(III) citrate and potassium ferricyanide, which is then dried in the dark. [4] When exposed to ultraviolet light, the chemicals react to form an insoluble blue dye, while the unexposed areas remain white after being washed with water. [5] Herschel's invention eventually became highly popular among architects and engineers in the twentieth century for reproducing technical drawings, which is how the term 'blueprint' originated.

The writer is considering deleting Sentence 2 from the passage. Should the writer make this deletion?

Show answer & explanation

Answer: No, because the sentence explains the original purpose that motivated Herschel to develop the cyanotype process.

Answer

The sentence should not be deleted because it explains the original purpose that motivated Herschel to develop the cyanotype process.
The sentence should be kept because it explains the original purpose that motivated Herschel to develop the cyanotype process. Deleting this sentence would leave a gap in the narrative, as the reader would not understand why Herschel sought to invent this photographic process in the first place.

Step-by-Step Solution

1
Analyze Sentence 2 in the context of the paragraph to determine its role.
Sentence 2 explains that Herschel wanted to find a reliable way to copy his mathematical and astronomical notes without rewriting them.
This identifies the specific information that would be lost if the sentence were deleted.
2
Evaluate the relevance of this information to the overall passage.
The passage discusses the invention, mechanism, and legacy of the cyanotype. Explaining why Herschel developed the process directly supports the main idea of how the cyanotype came to be.
This determines whether the sentence should be kept (No) or deleted (Yes).
3
Compare the correct stance (No) with the provided reasons in the choices.
The option explaining that the sentence provides the motivating purpose behind the invention matches the logical function of the text.
This yields the final correct answer.

Key Concept

Evaluating the relevance of supporting details and structural cohesion when deciding whether to add or delete sentences.
Estimated Time:1m 0s
Question 1589Question →

A basket contains red apples and green apples in a ratio of 2:52:5. If there are 35 apples in the basket, how many green apples are in the basket?

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Answer: 25

Answer

25
To find the number of green apples, first determine the total number of parts in the ratio by adding the two components: 2+5=72 + 5 = 7 parts. Next, divide the total number of apples in the basket by the total parts to find the value of each part: 35÷7=535 \div 7 = 5 apples per part. Finally, multiply the number of parts representing green apples (55 parts) by the value of each part (55): 5×5=255 \times 5 = 25. Therefore, the correct quantity of green apples in the basket is 25.

Step-by-Step Solution

1
Find the total number of parts in the ratio.
2+5=72 + 5 = 7 parts
The ratio of red apples to green apples is 2:52:5, so the total quantity is divided into 77 equal parts.
2
Calculate the value of a single part of the ratio.
35÷7=535 \div 7 = 5 apples per part
Divide the total number of apples (3535) by the total number of parts (77) to find how many apples make up one part.
3
Multiply the value of a single part by the number of parts representing green apples.
5 parts×5 apples/part=255 \text{ parts} \times 5 \text{ apples/part} = 25 green apples
Since green apples represent 55 parts of the ratio, multiply 55 by the value of one part (55) to find the total number of green apples.

Key Concept

Solving word problems involving part-to-part ratios and totals by finding the unit value per ratio part.
Question 1590Question →

Read the following passage segment:

Although the new software was designed to streamline inventory management, several employees remained indifferent of the technology, preferring their traditional paper logs.

Which of the following options represents the correct idiomatic word choice to replace the underlined portion?

Show answer & explanation

Answer: indifferent to

Answer

indifferent to
The correct answer is the phrase containing 'indifferent to'. In standard English, the adjective 'indifferent' must be paired with the preposition 'to' when describing a lack of interest, concern, or sympathy toward something.

Step-by-Step Solution

1
Identify the underlined phrase and the word that requires a prepositional partner.
The word is the adjective 'indifferent'.
Different adjectives require specific prepositions to form correct idiomatic English expressions.
2
Determine which preposition standardly pairs with 'indifferent' to convey the meaning of showing no interest or concern.
The preposition 'to' pairs with 'indifferent'.
'Indifferent to' is the established idiomatic collocation in standard English usage.

Key Concept

Idiomatic Preposition Collocations
Estimated Time:45s
Question 1591Question →

A positive integer KK has exactly 1212 positive factors. The greatest common divisor of KK and 8484 is 66, and the least common multiple of KK and 2424 is 360360. What is the value of KK?

Show answer & explanation

Answer: 90

Answer

90
The prime factorizations are 84=22×3×784 = 2^2 \times 3 \times 7, 6=2×36 = 2 \times 3, 24=23×324 = 2^3 \times 3, and 360=23×32×5360 = 2^3 \times 3^2 \times 5. The condition gcd⁡(K,84)=6\gcd(K, 84) = 6 implies the power of 2 in KK is exactly 1, and the power of 3 is at least 1. The condition lcm(K,24)=360\text{lcm}(K, 24) = 360 implies the power of 3 in KK is exactly 2, and the power of 5 is exactly 1. Thus, KK must be 21×32×51=902^1 \times 3^2 \times 5^1 = 90. Testing 9090, it has (1+1)(2+1)(1+1)=12(1+1)(2+1)(1+1) = 12 positive factors, which perfectly satisfies all conditions.

Step-by-Step Solution

1
Find the prime factorization of the given numbers in the problem.
84=22×3×784 = 2^2 \times 3 \times 7, 6=2×36 = 2 \times 3, 24=23×324 = 2^3 \times 3, and 360=23×32×5360 = 2^3 \times 3^2 \times 5.
Expressing these numbers in terms of their prime components allows us to determine the constraints on the prime factorization of KK.
2
Analyze the greatest common divisor constraint gcd⁡(K,84)=6\gcd(K, 84) = 6.
The prime 2 must have an exponent of exactly 1 in the factorization of KK (since 8484 has 222^2 and the greatest common divisor has only 212^1). The prime 3 must have an exponent of at least 1 in KK. The prime 7 cannot be a factor of KK.
The greatest common divisor selects the minimum exponent for each shared prime factor.
3
Analyze the least common multiple constraint lcm(K,24)=360\text{lcm}(K, 24) = 360.
The prime 3 must have an exponent of exactly 2 in KK (since 2424 has 313^1 and the least common multiple has 323^2). The prime 5 must have an exponent of exactly 1 in KK (since 2424 has 505^0 and the least common multiple has 515^1). KK cannot contain any prime factors other than 2, 3, and 5.
The least common multiple selects the maximum exponent for each prime factor present in either number.
4
Combine the exponent constraints to determine KK and verify its factor count.
K=21×32×51=90K = 2^1 \times 3^2 \times 5^1 = 90. The number of positive factors of 9090 is (1+1)(2+1)(1+1)=2×3×2=12(1+1)(2+1)(1+1) = 2 \times 3 \times 2 = 12.
This unique configuration matches all given greatest common divisor and least common multiple prime power requirements and satisfies the factor count condition.

Key Concept

Prime Factorization, Greatest Common Divisor, Least Common Multiple, and Number of Factors

Alternative Method

Instead of analyzing prime factor exponents, check the given multiple-choice options. Test each option by checking if it has exactly 12 factors, a greatest common divisor of 6 with 84, and a least common multiple of 360 with 24. Only the value 90 meets all three criteria.
Estimated Time:2m 30s
Question 1592Question →

A party planner is preparing gift bags. She has a box containing 12 toy cars and another box containing 18 stickers. She wants to purchase the minimum number of boxes of each toy so that she has the exact same number of toy cars and stickers, with no items left over. If she puts all of these purchased items into the gift bags, what is the minimum total number of items (cars and stickers combined) she will have?

Show answer & explanation

Answer: 72

Answer

72
To find the minimum total number of items, we must first find the least common multiple (LCM) of the box sizes, 12 and 18. The prime factorizations are 12=22×312 = 2^2 \times 3 and 18=2×3218 = 2 \times 3^2. The LCM is 22×32=4×9=362^2 \times 3^2 = 4 \times 9 = 36. This means the planner must buy 36 toy cars (3 boxes of 12) and 36 stickers (2 boxes of 18) to have an equal amount of each. The minimum total number of items combined is 36+36=7236 + 36 = 72.

Step-by-Step Solution

1
Identify that the number of toy cars and stickers must be equal and must be a multiple of the box sizes (12 and 18). Find the least common multiple (LCM) of 12 and 18 to determine the minimum number of each item needed.
LCM(12, 18) = 36
Since the planner wants the minimum number of boxes and equal amounts of both items, the number of each item must be the smallest common multiple of the two box sizes.
2
Calculate the total number of items by adding the number of toy cars and stickers together.
36 + 36 = 72
The question asks for the total number of items combined, so we must add the 36 toy cars to the 36 stickers.

Key Concept

Least Common Multiple (LCM)
Estimated Time:1m 30s
Question 1593Question →

Passage

For centuries, the Khasi people of Meghalaya, India, have guided the aerial roots of *Ficus elastica* trees across rushing rivers, slowly weaving them into living bridges. Unlike standard wooden or steel structures, these organic bridges grow stronger over time and can easily withstand the region's extreme monsoon floods. [1] Consequently, the process of growing a living bridge requires immense patience, often taking up to 1515 years before the structure can support any pedestrian weight.

Question

Which choice provides the most logical transition from the preceding sentence to the sentence that follows?

Show answer & explanation

Answer: . However, the

Answer

The correct option is '. However, the' because it properly signals a contrast transition between the durability of the bridges and the long time it takes to grow them, while avoiding a comma splice by correctly separating the two independent clauses with a period.
The correct choice is the option that starts a new sentence with the contrast transition 'However'. This transition accurately reflects the relationship between the two sentences: the first sentence highlights a positive trait of the living bridges (their durability), while the second introduces a limitation (the fifteen-year growth period). Additionally, separating the independent clauses into two distinct sentences avoids a comma splice.

Step-by-Step Solution

1
Analyze the relationship between the two sentences separated by the transition.
The first sentence explains a major advantage of the living root bridges (their durability in floods). The second sentence introduces a disadvantage (the long time required to grow them). This indicates a relationship of contrast.
To select the correct transition word, you must identify the logical connection between the ideas in both sentences.
2
Evaluate the grammar and punctuation of the options that express contrast.
Both the option with ', however, the' and the option with '. However, the' establish contrast. However, the option with ', however, the' results in a comma splice because it joins two independent clauses with only a comma. The option starting a new sentence with '. However, the' is grammatically correct.
A transition must not only make logical sense but also conform to standard punctuation rules for joining independent clauses.

Key Concept

Selecting the correct transition word to establish a logical relationship of contrast between sentences, while using correct punctuation to avoid comma splices.
Estimated Time:1m 0s
Question 1594Question →

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

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order of expressions from least to greatest is the expression evaluating to −25-25, followed by the expression evaluating to −8-8, then the expression evaluating to −4-4, and finally the expression evaluating to 1616.
Evaluating each expression using the correct order of operations yields −25-25 for the expression containing the negative base exponent, −8-8 for the expression containing the absolute value, −4-4 for the expression containing the cubed term, and 1616 for the expression containing the squared term. Ordering these values from least to greatest results in the sequence −25<−8<−4<16-25 < -8 < -4 < 16.

Step-by-Step Solution

1
Evaluate the first expression: 2×(−3)2−42÷82 \times (-3)^2 - 4^2 \div 8.
Value is 1616.
Exponents are calculated first: (−3)2=9(-3)^2 = 9 and 42=164^2 = 16. Then, perform multiplication and division: 2×9=182 \times 9 = 18 and 16÷8=216 \div 8 = 2. Finally, subtract: 18−2=1618 - 2 = 16.
2
Evaluate the second expression: −24+3×(5−8)-2^4 + 3 \times (5 - 8).
Value is −25-25.
Parentheses are evaluated first: 5−8=−35 - 8 = -3. Next, evaluate the exponent: −24=−16-2^4 = -16 (since the exponent applies to the base 22 before negation). Perform multiplication: 3×(−3)=−93 \times (-3) = -9. Finally, add: −16+(−9)=−25-16 + (-9) = -25.
3
Evaluate the third expression: (3−5)3÷4−2(3 - 5)^3 \div 4 - 2.
Value is −4-4.
Parentheses are evaluated first: 3−5=−23 - 5 = -2. Next, cube the result: (−2)3=−8(-2)^3 = -8. Perform division: −8÷4=−2-8 \div 4 = -2. Finally, subtract: −2−2=−4-2 - 2 = -4.
4
Evaluate the fourth expression: −∣−6−22×3∣+10-| -6 - 2^2 \times 3 | + 10.
Value is −8-8.
Within the absolute value bars, evaluate the exponent first: 22=42^2 = 4, then multiply: 4×3=124 \times 3 = 12. Subtract to find the absolute value interior: −6−12=−18-6 - 12 = -18. The absolute value of −18-18 is 1818, which is negated to −18-18. Finally, add 1010 to get −18+10=−8-18 + 10 = -8.
5
Order the final values from least to greatest.
−25<−8<−4<16-25 < -8 < -4 < 16.
Comparing the values −25-25, −8-8, −4-4, and 1616 on the number line determines their relative size.

Key Concept

Order of operations rules dictate that expressions are evaluated in the sequence of Parentheses/Grouping symbols, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Unary negation is performed after exponentiation unless grouped by parentheses.
Estimated Time:1m 30s
Question 1595Question →

An expression is shown below.

23×22−4×(1−3)3∣2−23∣−10 \frac{2^3 \times 2^2 - 4 \times (1 - 3)^3}{|2 - 2^3| - 10}

What is the value of this expression when it is simplified using the standard order of operations?

Show answer & explanation

Answer: -16

Answer

The correct answer is −16-16.
To find the value of the expression, follow the standard order of operations (PEMDAS). First, simplify terms with exponents and parentheses: 23=82^3 = 8, 22=42^2 = 4, and (1−3)3=(−2)3=−8(1 - 3)^3 = (-2)^3 = -8. The numerator becomes 8×4−4×(−8)=32−(−32)=648 \times 4 - 4 \times (-8) = 32 - (-32) = 64. In the denominator, evaluate the exponent first to get 2−8=−62 - 8 = -6, then apply the absolute value to get ∣−6∣=6|-6| = 6, and finally subtract 1010 to get 6−10=−46 - 10 = -4. Dividing the numerator by the denominator gives 64−4=−16\frac{64}{-4} = -16.

Step-by-Step Solution

1
Evaluate the exponents in the numerator and the denominator, including the term inside the parenthesis.
In the numerator, 23=82^3 = 8, 22=42^2 = 4, and (1−3)3=(−2)3=−8(1 - 3)^3 = (-2)^3 = -8. In the denominator, 23=82^3 = 8.
According to the order of operations (PEMDAS), operations inside grouping symbols (parentheses) and exponentiation must be performed before multiplication, division, addition, or subtraction.
2
Perform the multiplications in the numerator and evaluate the expression inside the absolute value in the denominator.
The numerator becomes 8×4−4×(−8)=32−(−32)8 \times 4 - 4 \times (-8) = 32 - (-32). The denominator becomes ∣2−8∣−10=∣−6∣−10|2 - 8| - 10 = |-6| - 10.
Multiplication has priority over addition and subtraction. In the denominator, the expression inside the absolute value acts as a grouping symbol and is simplified first.
3
Perform addition and subtraction in both the numerator and the denominator, including applying the absolute value.
The numerator is 32+32=6432 + 32 = 64. The denominator is 6−10=−46 - 10 = -4.
Subtracting a negative number is equivalent to adding its positive counterpart: 32−(−32)=32+3232 - (-32) = 32 + 32. The absolute value of −6-6 is 66, so the denominator simplifies to 6−10=−46 - 10 = -4.
4
Divide the simplified numerator by the simplified denominator.
64−4=−16\frac{64}{-4} = -16.
The fraction bar represents division, which is the final operation to perform to find the value of the expression.

Key Concept

Order of Operations and Number Properties

Alternative Method

Instead of evaluating 23×222^3 \times 2^2 separately, the laws of exponents can be used to simplify the term first: 23×22=23+2=25=322^3 \times 2^2 = 2^{3+2} = 2^5 = 32.
Estimated Time:1m 30s
Question 1596Question →

A writer is editing a draft of an essay about the history of timekeeping in the United States.

[1] In the mid-nineteenth century, the expansion of the American railroad network created an urgent need for standardized timekeeping. [2] Before this, towns set their clocks according to the sun's position at local noon, resulting in thousands of localized time zones across the country. [3] A passenger traveling from Boston to Washington, D.C., for example, had to adjust their watch dozens of times during the journey. [4] Standardized time zones were officially adopted in 1883, dividing the country into four distinct zones. [5] Railroad companies were privately owned and highly competitive during this era.

Which of the following represents the most logical decision regarding the proposed deletion of Sentence 5?

Show answer & explanation

Answer: Deleted, because it introduces a detail about the ownership and competitiveness of railroads that is tangential to the paragraph's focus on time standardization.

Answer

Deleted, because it introduces a detail about the ownership and competitiveness of railroads that is tangential to the paragraph's focus on time standardization.
The correct option is the one stating the sentence should be deleted because it introduces details about corporate ownership and competition that are tangential to the paragraph's core focus. The paragraph focuses strictly on the transition from localized solar time to standardized time zones due to railroad travel. Sentence 5 interrupts this progression with unrelated business history.

Step-by-Step Solution

1
Identify the primary topic and focus of the paragraph.
The paragraph describes the history of timekeeping in the United States, specifically the shift from local solar time to standardized time zones driven by the expansion of the railroad network.
Establishing the paragraph's main theme is necessary to evaluate the relevance of individual sentences.
2
Analyze the content and function of the sentence proposed for deletion.
Sentence 5 mentions that railroad companies were privately owned and highly competitive. This detail focuses on the business landscape of railroads rather than timekeeping.
Evaluating the specific contribution of the sentence helps determine if it supports the main topic or introduces a distraction.
3
Select the option that correctly recommends deleting the sentence and provides the accurate justification.
The decision to delete the sentence is correct because the sentence is tangential to the paragraph's main focus on time standardization, which aligns with the correct option's rationale.
Comparing the logical analysis with the given options ensures the choice with the correct decision and explanation is selected.

Key Concept

Evaluating the relevance of a sentence to the paragraph's main idea and deciding whether to retain or delete it based on passage unity.
Question 1597Question →

What is the value of the expression below?

−[(−3)2−∣4−2×5∣]×23−18÷3×(−2)−3+1 -[(-3)^2 - |4 - 2 \times 5|] \times 2^3 - \frac{18 \div 3 \times (-2)}{-3 + 1}
Show answer & explanation

Answer: -30

Answer

The value of the expression is −30-30.
Applying the correct order of operations (PEMDAS) ensures the expression is evaluated systematically. Evaluating the first term gives −[(−3)2−∣4−2×5∣]×23=−[9−∣−6∣]×8=−[9−6]×8=−3×8=−24-[(-3)^2 - |4 - 2 \times 5|] \times 2^3 = -[9 - |-6|] \times 8 = -[9 - 6] \times 8 = -3 \times 8 = -24. Evaluating the fraction term gives 18÷3×(−2)−3+1=6×(−2)−2=−12−2=6\frac{18 \div 3 \times (-2)}{-3 + 1} = \frac{6 \times (-2)}{-2} = \frac{-12}{-2} = 6. Subtracting the fraction value from the first term results in −24−6=−30-24 - 6 = -30.

Step-by-Step Solution

1
Simplify the grouping symbols: the bracket and absolute value term.
−3-3
First, evaluate the exponent (−3)2=9(-3)^2 = 9. Inside the absolute value, multiplication takes precedence over subtraction: 4−2×5=4−10=−64 - 2 \times 5 = 4 - 10 = -6. The absolute value is ∣−6∣=6|-6| = 6. Subtracting this from 99 inside the brackets gives 33. Finally, apply the negative sign in front of the brackets to get −3-3.
2
Evaluate the exponent 232^3 and perform the multiplication.
−24-24
Exponents are evaluated before multiplication. Since 23=82^3 = 8, the multiplication becomes −3×8=−24-3 \times 8 = -24.
3
Evaluate the fraction term.
66
In the numerator, multiplication and division are performed from left to right: 18÷3=618 \div 3 = 6, then 6×(−2)=−126 \times (-2) = -12. The denominator evaluates to −3+1=−2-3 + 1 = -2. Dividing the numerator by the denominator gives −12−2=6\frac{-12}{-2} = 6.
4
Perform the final subtraction.
−30-30
Subtract the evaluated fraction value from the first term: −24−6=−30-24 - 6 = -30.

Key Concept

Order of operations (PEMDAS), absolute value, and exponent rules with signed numbers.

Alternative Method

Evaluate each major term separately, paying close attention to grouping symbols, negative signs, and the left-to-right order for multiplication and division.
Estimated Time:2m 30s
Question 1598Question →

Early motion pictures were recorded on cellulose nitrate film, a material that was both highly flammable and chemically unstable. Over time, these historical reels undergo a process of decay, releasing acidic gases that degrade the emulsion until the image is completely lost. To halt this deterioration, archivist crews laboring in specialized vaults have to check out the films on a regular basis to monitor their condition. If a reel shows signs of decay, it is immediately copied onto a modern, stable polyester base.

Which choice most effectively maintains the established tone of the passage and is most concise?

Show answer & explanation

Answer: inspect the films periodically

Answer

inspect the films periodically
The correct choice is 'inspect the films periodically' because it preserves the formal, academic tone of the passage through appropriate vocabulary ('inspect', 'periodically') and maintains optimal conciseness by avoiding redundant modifiers and wordy prepositional phrases.

Step-by-Step Solution

1
Analyze the context and established register of the passage.
The surrounding text uses formal, precise, and objective language appropriate for a scientific or historical exposition (e.g., 'cellulose nitrate', 'chemically unstable', 'degrade the emulsion').
Identifying the target tone is necessary to ensure the replaced phrase maintains stylistic consistency.
2
Identify the stylistic flaw in the underlined text.
The phrase 'have to check out' is conversational and informal, causing a clear tone clash with the rest of the academic passage.
Recognizing the tone register mismatch determines why the original text must be changed.
3
Evaluate the choices for tone, precision, and conciseness.
The option to 'inspect the films periodically' is formal and concise. The option to 'conduct meticulous examinations...' is too pretentious and wordy, while the option to 'regularly inspect the films over and over again' contains redundant frequency terms.
Eliminating options that are overly informal, overly formal, or redundant leaves the single correct, stylistically balanced choice.

Key Concept

Stylistic Appropriateness
Estimated Time:1m 0s
Question 1599Question →

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

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order from least to greatest is: 6.8×10−56.8 \times 10^{-5}, 7.1×10−47.1 \times 10^{-4}, 1.5×10−31.5 \times 10^{-3}, and 2.4×10−32.4 \times 10^{-3}.
The correct order is determined by first arranging the values by their power of 10 exponent from least to greatest: −5<−4<−3-5 < -4 < -3. This places 6.8×10−56.8 \times 10^{-5} as the smallest and 7.1×10−47.1 \times 10^{-4} as the second smallest. For the remaining two values that share the exponent −3-3, we compare their coefficients: 1.5<2.41.5 < 2.4, placing 1.5×10−31.5 \times 10^{-3} before 2.4×10−32.4 \times 10^{-3}.

Step-by-Step Solution

1
Compare the exponents of the base 10 to establish the primary order of magnitude.
The exponents are −5-5, −4-4, and −3-3. Since −5<−4<−3-5 < -4 < -3, we know that any number with an exponent of −5-5 is smaller than one with −4-4, which in turn is smaller than one with −3-3.
In scientific notation, a smaller exponent on the power of 10 indicates a smaller overall value because it represents shifting the decimal point further to the left.
2
Place the numbers with unique exponents in order.
The smallest number is 6.8×10−56.8 \times 10^{-5}, followed by 7.1×10−47.1 \times 10^{-4}.
These have the smallest exponents (−5-5 and −4-4, respectively).
3
Compare the coefficients of the remaining numbers that share the same exponent.
Both 1.5×10−31.5 \times 10^{-3} and 2.4×10−32.4 \times 10^{-3} share the exponent −3-3. Comparing their coefficients, 1.5<2.41.5 < 2.4, so 1.5×10−3<2.4×10−31.5 \times 10^{-3} < 2.4 \times 10^{-3}.
When the powers of 10 are identical, the values can be compared directly by their coefficients.

Key Concept

Comparing and ordering numbers written in scientific notation by analyzing their powers of 10 and coefficients.
Question 1600Question →

Venus flytraps are carnivorous plants that capture their prey using specialized leaves. When an insect touches the sensitive trigger hairs on the inner surface of the trap, the leaves snap shut in a fraction of a second. [1] The rapid closure of the trap, which happens in a very fast and quick manner, prevents the insect from escaping. The plant then secretes digestive enzymes to break down its meal.

Which choice most effectively and concisely completes the sentence?

Show answer & explanation

Answer: The rapid closure of the trap

Answer

The correct answer is the choice that states 'The rapid closure of the trap'.
The correct option is the most concise and direct way to convey the meaning of the sentence. It avoids redundant descriptors since the adjective 'rapid' already conveys the speed of the closure.

Step-by-Step Solution

1
Identify redundant and wordy elements in the underlined portion.
The original phrase includes the synonyms 'rapid', 'fast', and 'quick' in a short span, which is redundant.
ACT English prioritizes concise and direct phrasing that avoids repeating the same idea with different words.
2
Evaluate the remaining choices to ensure they are grammatically correct and preserve meaning.
One option resolves the redundancy but introduces a subordinating structure that creates a sentence fragment. Another option introduces incorrect punctuation around a restrictive modifier. The best option removes all redundant terms while leaving the sentence structure grammatically sound.
The correct choice must improve conciseness without introducing grammatical errors like fragments or improper comma usage.

Key Concept

Eliminating redundancy and maintaining grammatical structure by choosing the most concise correct phrasing.
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