Exponents & Roots — Practice Set

Bank code: ARI-EXP · Section: Arithmetic · 30 questions — Easy 5 · Medium 8 · Hard 12 · Extreme 5

Exponent questions are almost never about computing a big number - match the bases, factor instead of adding across a plus sign, and remember that x^2 = k has two roots while sqrt(k) has one.

Every question below is live in the question bank under the ID shown — the sheet and the portal are the same questions. Attempt a level with the clock running, then check Part C.

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Part A — Questions

Quantitative Comparison — the four choices are always the same, so they are not reprinted: (A) Column A is greater · (B) Column B is greater · (C) The two quantities are equal · (D) The relationship cannot be determined from the information given

Level 1 · Easy — 5 questions · ~4 min

warm-up — these must be automatic

Q1 · ARI-EXP-006 · QC · 45s

Column A: (-2)^4 Column B: -2^4

Q2 · ARI-EXP-008 · MCQ · 45s

Which of the following is equal to sqrt(75)?

(A) 3 sqrt(5)
(B) 5 sqrt(3)
(C) 5 sqrt(15)
(D) 25 sqrt(3)
(E) 37.5

Q3 · ARI-EXP-011 · MCQ · 55s

What is the value of (2^8 x 2^(-3)) / 2^2 ?

(A) -512
(B) 8
(C) 32
(D) 128
(E) 512

Q4 · ARI-EXP-013 · Select all that apply · 60s

Which of the following are equal to 32? Select all that apply.

(A) 2^5
(B) 4^3 / 2
(C) 2^3 x 2^2
(D) (2^2)^3
(E) (-2)^5

Q5 · ARI-EXP-015 · Numeric Entry · 40s

What is the value of (1/3)^(-2)? Enter your answer as a number.

Numeric entry — write the number.

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Level 2 · Medium — 8 questions · ~9 min

two or three steps, one planted trap each

Q6 · ARI-EXP-021 · MCQ · 60s

If 2^x = 32, what is the value of 2^(x+3)?

(A) 35
(B) 96
(C) 160
(D) 256
(E) 32,768

Q7 · ARI-EXP-025 · MCQ · 75s

Which of the following is equal to sqrt(8)/sqrt(6), with no radical in the denominator?

(A) 2/3
(B) sqrt(3)/3
(C) 2 sqrt(3)/3
(D) 4/3
(E) 2 sqrt(3)

Q8 · ARI-EXP-026 · QC · 60s

0 < x < 1

Column A: x^3 Column B: x^2

Q9 · ARI-EXP-037 · Select all that apply · 80s

Which of the following are rational numbers? Select all that apply.

(A) sqrt(49)
(B) sqrt(0.4)
(C) the cube root of -8
(D) sqrt(50)
(E) sqrt(3) x sqrt(12)

Q10 · ARI-EXP-039 · Numeric Entry · 65s

What is the value of 16^(3/4)? Enter your answer as a number.

Numeric entry — write the number.

Q11 · ARI-EXP-043 · Numeric Entry · 75s

The expression sqrt(45) + sqrt(80) - sqrt(20) can be written as k x sqrt(5) for some integer k. What is k? Enter your answer as a number.

Numeric entry — write the number.

Q12 · ARI-EXP-045 · MCQ · 65s

Which of the following is equal to (2x^3 y^2)^3 ?

(A) 2x^9 y^6
(B) 6x^6 y^5
(C) 6x^9 y^6
(D) 8x^6 y^5
(E) 8x^9 y^6

Q13 · ARI-EXP-049 · QC · 85s

Column A: (1/2)^100 Column B: (1/3)^70

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Level 3 · Hard — 12 questions · ~18 min

where 162+ is won or lost

Q14 · ARI-EXP-042 · MCQ · 100s

Which of the following is the greatest?

(A) 2^25
(B) 3^20
(C) 4^15
(D) 6^10
(E) 12^5

Q15 · ARI-EXP-052 · QC · 80s

x is a nonzero real number.

Column A: x^2 Column B: x^3

Q16 · ARI-EXP-054 · MCQ · 85s

If x > 0 and x^(2/3) = 4, what is x?

(A) 2
(B) 6
(C) 8
(D) 16
(E) 64

Q17 · ARI-EXP-055 · Select all that apply · 95s

a and b are nonzero real numbers. Which of the following are equal to sqrt(a^2 b^4) for ALL such a and b? Select all that apply.

(A) a b^2
(B) |a| b^2
(C) |a b^2|
(D) sqrt(a^2) x sqrt(b^4)
(E) |a| |b|^4

Q18 · ARI-EXP-057 · MCQ · 80s

n is an integer. For which values of n is (-1)^n + (-1)^(n+1) = 0 ?

(A) Even n only
(B) Odd n only
(C) All integers n
(D) No integers n
(E) Only n = 0

Q19 · ARI-EXP-059 · MCQ · 90s

Which of the following is equal to 3/(2 + sqrt(5)) with a rational denominator?

(A) 3 sqrt(5) - 6
(B) 6 - 3 sqrt(5)
(C) 3(2 - sqrt(5))/9
(D) 3(2 + sqrt(5))/9
(E) 3/2 + 3/sqrt(5)

Q20 · ARI-EXP-060 · QC · 85s

x > 0

Column A: x^(1/2) Column B: x^(1/3)

Q21 · ARI-EXP-061 · MCQ · 80s

What is the value of (sqrt(5) - sqrt(3))^2 ?

(A) 2
(B) 8
(C) 8 - 2 sqrt(8)
(D) 8 - 2 sqrt(15)
(E) 8 + 2 sqrt(15)

Q22 · ARI-EXP-062 · Numeric Entry · 85s

If 3^x = 5, what is the value of 9^(x+1)? Enter your answer as a number.

Numeric entry — write the number.

Q23 · ARI-EXP-063 · Select all that apply · 95s

x and y are positive numbers and a and b are nonzero real numbers. Which of the following are true for all such values? Select all that apply.

(A) x^a x x^b = x^(a+b)
(B) (x + y)^2 = x^2 + y^2
(C) sqrt(x^2 + y^2) = x + y
(D) x^a / x^b = x^(a-b)
(E) sqrt(xy) = sqrt(x) x sqrt(y)

Q24 · ARI-EXP-065 · MCQ · 95s

What is the value of 8^(2/3) + 16^(3/4) + 81^(1/4) - 32^(1/5) ?

(A) 5
(B) 7
(C) 11
(D) 13
(E) 19

Q25 · ARI-EXP-066 · Numeric Entry · 90s

If 2^x + 2^x + 2^x + 2^x = 4^12, what is x? Enter your answer as a number.

Numeric entry — write the number.

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Level 4 · Extreme — 5 questions · ~10 min

165+ — expect to need the insight, not the grind

Q26 · ARI-EXP-073 · Numeric Entry · 110s

If 2^x = 3, 3^y = 4, and 4^z = 5, what is the value of 2^(xyz)? Enter your answer as a number.

Numeric entry — write the number.

Q27 · ARI-EXP-076 · Select all that apply · 120s

n is a positive integer. Which of the following expressions are integers for EVERY such n? Select all that apply.

(A) n^(1/n)
(B) (2^n + 2^n)^(1/n)
(C) (n^n)^(1/n)
(D) (n^2)^(1/2)
(E) (9^n)^(1/(2n))

Q28 · ARI-EXP-077 · Numeric Entry · 115s

What is the units digit of 3^47 + 7^35? Enter your answer as a number.

Numeric entry — write the number.

Q29 · ARI-EXP-078 · MCQ · 120s

x is a positive number and x + 1/x = sqrt(5). What is the value of x^4 + 1/x^4 ?

(A) 3
(B) 5
(C) 7
(D) 9
(E) 25

Q30 · ARI-EXP-079 · QC · 115s

Column A: sqrt(6 + sqrt(6 + sqrt(6 + ...))), where the nesting continues without end Column B: 3

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