ACT Math Practice Test: Exponents, Radicals and Polynomials
- Exponents, Radicals and Polynomials
- 10 questions
- 12 minutes
These 10 questions practice the algebra skills behind many ACT Math items: exponent rules, radicals, polynomial operations and factoring. When an equation involves a square root, always check your solutions in the original equation, because squaring can create answers that do not work.
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Question 1
For all x ≠ 0, which expression is equivalent to (x3)2 · x4x5?
Correct answer: B
A power of a power multiplies exponents: (x³)² = x⁶. Then x⁶ · x⁴ = x¹⁰, and dividing by x⁵ gives x⁵. Adding 3 + 2 instead of multiplying leads to x⁴.
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Question 2
Which of the following is equal to 72?
Correct answer: D
72 = 36 × 2, and √36 = 6, so √72 = 6√2. Check the others by squaring: (2√6)² = 24, (4√3)² = 48 and (3√6)² = 54, none of which is 72.
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Question 3
What is the value of 1634?
Correct answer: A
The fourth root of 16 is 2, and 2³ = 8. The choice 12 multiplies 16 by 3/4, which treats the exponent as a factor.
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Question 4
Which expression is equivalent to (2x − 3)2?
Correct answer: C
(2x − 3)(2x − 3) = 4x² − 6x − 6x + 9 = 4x² − 12x + 9. Squaring only the two terms gives 4x² + 9 and misses the middle term.
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Question 5
What is the remainder when x3 − 2x2 + 4x − 5 is divided by x − 2?
Correct answer: B
By the remainder theorem, the remainder is the value of the polynomial at x = 2: 8 − 8 + 8 − 5 = 3. Using x = −2 gives −29, a sign error.
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Question 6
Which of the following is the complete factorization of 3x3 − 12x?
Correct answer: A
Take out the common factor 3x: 3x(x² − 4). Then x² − 4 is a difference of squares, (x − 2)(x + 2). The choice with (x − 4)(x + 4) would multiply out to 3x³ − 48x.
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Question 7
What is the value of (5 + 2)(5 − 2)?
Correct answer: C
This is a difference of squares: (√5)² − 2² = 5 − 4 = 1. Adding the squares instead of subtracting gives 9.
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Question 8
Which of the following is equal to 63?
Correct answer: D
Multiply the top and bottom by √3: 6√3 ÷ 3 = 2√3. The choice 6√3 multiplies the top by √3 but forgets to divide by the new denominator, 3.
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Question 9
Which expression is equivalent to (3x2 − 5x + 2) − (x2 + 2x − 7)?
Correct answer: B
Distribute the minus sign to every term in the second polynomial: 3x² − 5x + 2 − x² − 2x + 7 = 2x² − 7x + 9. The choice ending in −5 subtracts only the first term and adds the rest.
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Question 10
What is the solution set of the equation x + 7 = x − 5?
Correct answer: A
Squaring gives x + 7 = x² − 10x + 25, so x² − 11x + 18 = 0 and x = 2 or x = 9. Check: for x = 9, √16 = 4 = 9 − 5. For x = 2, √9 = 3 but 2 − 5 = −3, so 2 is extraneous.
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