ACCUPLACER Quantitative Reasoning, Algebra & Statistics Quiz: Integer Exponents
20 questions · exam conditions
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Integer ExponentsQuestion 1 of 20

Which of the following is equivalent to the expression (34)−2\left(\frac{3}{4}\right)^{-2}?

−169-\frac{16}{9}
−916-\frac{9}{16}
916\frac{9}{16}
169\frac{16}{9}
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ACCUPLACER Quantitative Reasoning, Algebra & Statistics Quiz

ACCUPLACER Quantitative Reasoning, Algebra & Statistics Quiz: Integer Exponents

Practice Integer Exponents in ACCUPLACER Quantitative Reasoning, Algebra & Statistics with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Integer Exponents, giving you a quick way to practice the rules, question types, and explanations that matter most for ACCUPLACER Quantitative Reasoning, Algebra & Statistics.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Which of the following is equivalent to the expression (34)−2\left(\frac{3}{4}\right)^{-2}?

  1. −169-\frac{16}{9}
  2. −916-\frac{9}{16}
  3. 916\frac{9}{16}
  4. 169\frac{16}{9} (correct answer)
Explanation: A negative exponent indicates taking the reciprocal of the base. Therefore, (34)−2=(43)2\left(\frac{3}{4}\right)^{-2} = \left(\frac{4}{3}\right)^{2}. Squaring the fraction means squaring both the numerator and the denominator: 4232=169\frac{4^2}{3^2} = \frac{16}{9}.
  • A is incorrect because the negative exponent does not make the result negative, and the fraction is inverted incorrectly.
  • B is incorrect because the negative exponent means to take the reciprocal of the base, not to make the result negative.
  • C is incorrect because it squares the original fraction without taking its reciprocal first, which is what the negative exponent requires.

Question 2

What is the value of 2−1+4−12^{-1} + 4^{-1}?

  1. −12-\frac{1}{2}
  2. 16\frac{1}{6}
  3. 13\frac{1}{3}
  4. 34\frac{3}{4} (correct answer)
Explanation: The expression must be evaluated term by term. The negative exponent indicates a reciprocal. So, 2−1=122^{-1} = \frac{1}{2} and 4−1=144^{-1} = \frac{1}{4}. The problem becomes 12+14\frac{1}{2} + \frac{1}{4}. To add these fractions, find a common denominator, which is 4. 12=24\frac{1}{2} = \frac{2}{4}. So, the sum is 24+14=34\frac{2}{4} + \frac{1}{4} = \frac{3}{4}.
  • A is incorrect. This may result from misinterpreting negative exponents as making the numbers negative.
  • B is incorrect. This is a common error from adding the denominators (2+4=62+4=6) or from incorrectly adding the bases first, (2+4)−1=6−1=1/6(2+4)^{-1} = 6^{-1} = 1/6.
  • C is incorrect. This can result from the common fraction addition error of adding numerators and denominators: 1+12+4=26=13\frac{1+1}{2+4} = \frac{2}{6} = \frac{1}{3}.

Question 3

The number of cells in a culture triples every hour. If there are NN cells at 12:00 PM, which expression represents the number of cells that were in the culture at 9:00 AM on the same day?

  1. 3−3N3^{-3}N (correct answer)
  2. 33N3^3 N
  3. N/3N/3
  4. N−3N-3
Explanation: The time 9:00 AM is 3 hours before 12:00 PM. The population is modeled by N⋅3tN \cdot 3^t, where tt is the number of hours after 12:00 PM. To find the population 3 hours before 12:00 PM, we must use t=−3t = -3. So the expression is N⋅3−3N \cdot 3^{-3}, which can also be written as N33\frac{N}{3^3} or N27\frac{N}{27}.
  • B is incorrect. This expression represents the population 3 hours after 12:00 PM, at 3:00 PM.
  • C is incorrect. This represents the population one hour prior, at 11:00 AM, not three hours prior.
  • D is incorrect. This represents a linear decrease, but the cell growth is exponential.

Question 4

If x≠0x \neq 0, what is the value of the expression 4x0−(2x)04x^0 - (2x)^0?

  1. 0
  2. 2
  3. 3 (correct answer)
  4. 4
Explanation: Any non-zero quantity raised to the power of 0 is equal to 1. The expression has two terms. In the first term, 4x04x^0, only the xx is raised to the power of 0. So, 4x0=4(1)=44x^0 = 4(1) = 4. In the second term, (2x)0(2x)^0, the entire quantity 2x2x is raised to the power of 0, so (2x)0=1(2x)^0 = 1. The expression simplifies to 4−1=34 - 1 = 3.
  • A is incorrect. This result might come from incorrectly evaluating 4x04x^0 as (4x)0=1(4x)^0=1, leading to 1−1=01-1=0.
  • B is incorrect. This result might come from misinterpreting the parentheses, as in 4x0−2x0=2x0=2(1)=24x^0 - 2x^0 = 2x^0 = 2(1) = 2.
  • D is incorrect. This would happen if the second term, (2x)0(2x)^0, was incorrectly evaluated as 0.

Question 5

If a=−3a = -3, what is the value of the expression −a2−a3-a^2 - a^3?

  1. -36
  2. -18
  3. 18 (correct answer)
  4. 36
Explanation: Substitute a=−3a = -3 into the expression: −(−3)2−(−3)3-(-3)^2 - (-3)^3. Following the order of operations, evaluate the exponents first. For the first term, (−3)2=(−3)(−3)=9(-3)^2 = (-3)(-3) = 9. The term becomes −(9)-(9), or −9-9. For the second term, (−3)3=(−3)(−3)(−3)=−27(-3)^3 = (-3)(-3)(-3) = -27. The expression is now −9−(−27)-9 - (-27), which simplifies to −9+27=18-9 + 27 = 18.
  • A is incorrect. This results from incorrectly evaluating (−3)3(-3)^3 as 2727, leading to −9−27=−36-9 - 27 = -36.
  • B is incorrect. This results from making sign errors on both terms, possibly calculating −(−3)2-(-3)^2 as 99 and (−3)3(-3)^3 as 2727, leading to 9−27=−189-27=-18.
  • D is incorrect. This results from a common error in the first term, calculating −a2-a^2 as (−a)2(-a)^2, which would be (−(−3))2=32=9(-(-3))^2 = 3^2 = 9. This leads to 9−(−27)=369 - (-27) = 36.

Question 6

Which of the following expressions is equivalent to (3x4)−2(3x^4)^{-2} for x≠0x \neq 0?

  1. 19x8\frac{1}{9x^8} (correct answer)
  2. 3x8\frac{3}{x^8}
  3. 9x8\frac{9}{x^8}
  4. −9x8-9x^8
Explanation: The exponent −2-2 applies to both the coefficient 3 and the variable part x4x^4. So, (3x4)−2=3−2(x4)−2(3x^4)^{-2} = 3^{-2}(x^4)^{-2}. For the coefficient, 3−2=132=193^{-2} = \frac{1}{3^2} = \frac{1}{9}. For the variable, use the power of a power rule: (x4)−2=x4⋅−2=x−8(x^4)^{-2} = x^{4 \cdot -2} = x^{-8}. Since x−8=1x8x^{-8} = \frac{1}{x^8}, combining the parts gives 19⋅1x8=19x8\frac{1}{9} \cdot \frac{1}{x^8} = \frac{1}{9x^8}.
  • B is incorrect. This is a common error where the exponent −2-2 is applied to x4x^4 but not to the coefficient 3.
  • C is incorrect. This happens if the coefficient 3 is squared, but the negative sign on the exponent (which indicates a reciprocal) is ignored for the coefficient.
  • D is incorrect. This reflects multiple errors, including multiplying the coefficient by the exponent and incorrectly handling the signs.

Question 7

What is the value of (102−100)−1(10^2 - 10^0)^{-1}?

  1. −99100-\frac{99}{100}
  2. 1100\frac{1}{100}
  3. 199\frac{1}{99} (correct answer)
  4. 190\frac{1}{90}
Explanation: Following the order of operations, evaluate the expression inside the parentheses first. 102=10010^2 = 100 and 100=110^0 = 1. So, the expression inside the parentheses is 100−1=99100 - 1 = 99. The problem then becomes 99−199^{-1}. A negative exponent indicates a reciprocal, so 99−1=19999^{-1} = \frac{1}{99}.
  • A is incorrect. This results from the common mistake of incorrectly distributing the exponent: (102)−1−(100)−1=10−2−1=1100−1=−99100(10^2)^{-1} - (10^0)^{-1} = 10^{-2} - 1 = \frac{1}{100} - 1 = -\frac{99}{100}.
  • B is incorrect. This results from incorrectly evaluating 10010^0 as 0, which gives (100−0)−1=100−1=1100(100-0)^{-1} = 100^{-1} = \frac{1}{100}.
  • D is incorrect. This could result from misreading 10010^0 as 10110^1, which would lead to (100−10)−1=90−1=190(100-10)^{-1} = 90^{-1} = \frac{1}{90}.

Question 8

The expression (xa)2(x3)−1(x^a)^2(x^3)^{-1} simplifies to x7x^7. What is the value of aa?

  1. 2
  2. 4
  3. 5 (correct answer)
  4. 8
Explanation: First, simplify the left side of the equation using exponent rules. (xa)2=x2a(x^a)^2 = x^{2a} and (x3)−1=x−3(x^3)^{-1} = x^{-3}. Multiplying these gives x2a⋅x−3=x2a−3x^{2a} \cdot x^{-3} = x^{2a-3}. We are given that this simplifies to x7x^7. Therefore, the exponents must be equal: 2a−3=72a - 3 = 7. To solve for aa, add 3 to both sides: 2a=102a = 10. Then, divide by 2: a=5a = 5.
  • A is incorrect. If a=2a=2, the exponent would be 2(2)−3=12(2)-3=1.
  • B is incorrect. If a=4a=4, the exponent would be 2(4)−3=52(4)-3=5.
  • D is incorrect. This could result from an error in solving the equation, such as 2a=10→a=82a=10 \rightarrow a=8, or from setting up the equation incorrectly, like 2a+3=72a+3=7 which gives 2a=4,a=22a=4, a=2.

Question 9

Which of the following values is equal to 12−5\frac{1}{2^{-5}}?

  1. -32
  2. -10
  3. 132\frac{1}{32}
  4. 32 (correct answer)
Explanation: The expression 1a−n\frac{1}{a^{-n}} is equivalent to ana^n. Therefore, 12−5\frac{1}{2^{-5}} is equal to 252^5. Calculating 252^5 gives 2⋅2⋅2⋅2⋅2=322 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32.
  • A is incorrect. This comes from incorrectly applying a negative sign to the result.
  • B is incorrect. This is a result of incorrectly multiplying the base by the exponent (2⋅5=102 \cdot 5 = 10) and making a sign error.
  • C is incorrect. This is the value of 2−52^{-5}, not 12−5\frac{1}{2^{-5}}.

Question 10

If 3x=813^x = 81 and 2y=642^y = 64, what is the value of x−yx-y?

  1. -2 (correct answer)
  2. -1
  3. 1
  4. 2
Explanation: First, find the value of xx. We need to determine what power of 3 equals 81. 31=33^1=3, 32=93^2=9, 33=273^3=27, 34=813^4=81. So, x=4x=4. Next, find the value of yy. We need to determine what power of 2 equals 64. 21=22^1=2, 22=42^2=4, 23=82^3=8, 24=162^4=16, 25=322^5=32, 26=642^6=64. So, y=6y=6. The question asks for the value of x−yx-y, which is 4−6=−24-6 = -2.
  • B is incorrect. This might result from a calculation error for xx or yy.
  • C is incorrect. This might result from a calculation error for xx or yy.
  • D is incorrect. This is the value of y−xy-x, not x−yx-y.

Question 11

What is the value of the expression 40−4−24^0 - 4^{-2}?

  1. −116-\frac{1}{16}
  2. 1516\frac{15}{16} (correct answer)
  3. 1716\frac{17}{16}
  4. 16
Explanation: First, evaluate each term. Any non-zero number raised to the power of 0 is 1, so 40=14^0 = 1. A negative exponent indicates a reciprocal, so 4−2=142=1164^{-2} = \frac{1}{4^2} = \frac{1}{16}. The expression becomes 1−1161 - \frac{1}{16}. To subtract, find a common denominator: 1=16161 = \frac{16}{16}. So, 1616−116=1516\frac{16}{16} - \frac{1}{16} = \frac{15}{16}.
  • A is incorrect. This results from incorrectly evaluating 404^0 as 0.
  • C is incorrect. This results from a sign error, calculating 1−(−116)=1+116=17161 - (-\frac{1}{16}) = 1 + \frac{1}{16} = \frac{17}{16}.
  • D is incorrect. This is a common mistake from misapplying exponent rules, treating the subtraction as if it were division, leading to 40−(−2)=42=164^{0 - (-2)} = 4^2 = 16.

Question 12

Which of the following expressions is equivalent to (2x3)2(x−1)(2x^3)^2(x^{-1}) for x≠0x \neq 0?

  1. 2x52x^5
  2. 4x54x^5 (correct answer)
  3. 4x64x^6
  4. 4x74x^7
Explanation: First, evaluate (2x3)2(2x^3)^2. The exponent 2 applies to both the coefficient 2 and the variable x3x^3. This gives 22(x3)2=4x3⋅2=4x62^2(x^3)^2 = 4x^{3 \cdot 2} = 4x^6. Now, multiply this result by x−1x^{-1}: 4x6⋅x−14x^6 \cdot x^{-1}. Using the product of powers rule (am⋅an=am+na^m \cdot a^n = a^{m+n}), we get 4x6+(−1)=4x54x^{6+(-1)} = 4x^5.
  • A is incorrect. This results from forgetting to square the coefficient 2.
  • C is incorrect. This is the intermediate result before multiplying by x−1x^{-1}.
  • D is incorrect. This results from incorrectly adding the exponents, perhaps by making a sign error: 4x6+1=4x74x^{6+1} = 4x^7.

Question 13

What is the value of the expression 5⋅10−25 \cdot 10^{-2}?

  1. 0.005
  2. 0.05 (correct answer)
  3. 500
  4. 512
Explanation: First, evaluate the exponential term. 10−2=1102=110010^{-2} = \frac{1}{10^2} = \frac{1}{100}, which is equal to the decimal 0.01. Then, multiply this value by 5: 5⋅0.01=0.055 \cdot 0.01 = 0.05.
  • A is incorrect. This would be the value of 5⋅10−35 \cdot 10^{-3}.
  • C is incorrect. This results from misinterpreting the negative exponent as positive and adding zeros, as in 5⋅102=5005 \cdot 10^2 = 500.
  • D is incorrect. This answer seems to come from an unrelated calculation and is highly implausible.

Question 14

Which of the following expressions is equivalent to (x4)3x2\frac{(x^4)^3}{x^2} for all x≠0x \neq 0?

  1. x5x^5
  2. x6x^6
  3. x10x^{10} (correct answer)
  4. x14x^{14}
Explanation: First, simplify the numerator. According to the power of a power rule, (am)n=amn(a^m)^n = a^{mn}, so (x4)3=x4⋅3=x12(x^4)^3 = x^{4 \cdot 3} = x^{12}. Now the expression is x12x2\frac{x^{12}}{x^2}. According to the quotient of powers rule, aman=am−n\frac{a^m}{a^n} = a^{m-n}, so x12x2=x12−2=x10\frac{x^{12}}{x^2} = x^{12-2} = x^{10}.
  • A is incorrect. This results from adding the exponents in the numerator (4+3=74+3=7) instead of multiplying, giving x7x2=x5\frac{x^7}{x^2} = x^5.
  • B is incorrect. This results from dividing the exponents (12÷2=612 \div 2 = 6) instead of subtracting.
  • D is incorrect. This results from subtracting the denominator's exponent from the numerator's exponents incorrectly, possibly by adding them, as in x12+2=x14x^{12+2} = x^{14}.

Question 15

Which of the following is equivalent to (x−3y2)−2\left(\frac{x^{-3}}{y^2}\right)^{-2} for x≠0,y≠0x \neq 0, y \neq 0?

  1. 1x5\frac{1}{x^5}
  2. y4x6\frac{y^4}{x^6}
  3. x6y4\frac{x^6}{y^4}
  4. x6y4x^6y^4 (correct answer)
Explanation: To simplify the expression, apply the outer exponent −2-2 to both the numerator and the denominator: (x−3)−2(y2)−2\frac{(x^{-3})^{-2}}{(y^2)^{-2}}. Using the power of a power rule ((am)n=amn(a^m)^n=a^{mn}), the numerator becomes x(−3)(−2)=x6x^{(-3)(-2)} = x^6 and the denominator becomes y(2)(−2)=y−4y^{(2)(-2)} = y^{-4}. The expression is x6y−4\frac{x^6}{y^{-4}}. Since a negative exponent in the denominator is equivalent to a positive exponent in the numerator, y−4=1y4y^{-4} = \frac{1}{y^4}, so x6y−4=x6y4\frac{x^6}{y^{-4}} = x^6y^4.
  • A is incorrect. This may result from incorrectly adding exponents, such as x−3−2=x−5x^{-3-2}=x^{-5} and y2−2=y0=1y^{2-2}=y^0=1.
  • B is incorrect. This results from correctly applying the outer exponent to the denominator but making a sign error on the numerator's exponent.
  • C is incorrect. This results from correctly simplifying the numerator but making a sign error on the denominator's exponent.

Question 16

Which of the following is equivalent to 2a−3b46a2b−1\frac{2a^{-3}b^4}{6a^2b^{-1}} for a≠0,b≠0a \neq 0, b \neq 0?

  1. b33a\frac{b^3}{3a}
  2. b33a5\frac{b^3}{3a^5}
  3. b53a5\frac{b^5}{3a^5} (correct answer)
  4. 3b5a5\frac{3b^5}{a^5}
Explanation: Simplify the expression in parts: coefficients, aa terms, and bb terms. The coefficients 26\frac{2}{6} simplify to 13\frac{1}{3}. For the variables, use the quotient rule xmxn=xm−n\frac{x^m}{x^n} = x^{m-n}. For aa, we have a−3a2=a−3−2=a−5\frac{a^{-3}}{a^2} = a^{-3-2} = a^{-5}. For bb, we have b4b−1=b4−(−1)=b4+1=b5\frac{b^4}{b^{-1}} = b^{4-(-1)} = b^{4+1} = b^5. Combining these parts gives 13a−5b5\frac{1}{3}a^{-5}b^5. Since a−5=1a5a^{-5} = \frac{1}{a^5}, the expression is b53a5\frac{b^5}{3a^5}.
  • A is incorrect. This results from adding exponents instead of subtracting (a−3+2=a−1a^{-3+2}=a^{-1}) and subtracting exponents incorrectly (b4−1=b3b^{4-1}=b^3).
  • B is incorrect. This results from subtracting exponents for bb incorrectly (b4−1=b3b^{4-1}=b^3).
  • D is incorrect. This results from an error in simplifying the coefficients (6/2=36/2=3 instead of 2/6=1/32/6=1/3).

Question 17

If y=2x−3y = 2x^{-3} and x=−2x = -2, what is the value of yy?

  1. -16
  2. −14-\frac{1}{4} (correct answer)
  3. 14\frac{1}{4}
  4. 12
Explanation: First, substitute x=−2x = -2 into the expression for yy: y=2(−2)−3y = 2(-2)^{-3}. According to the order of operations, evaluate the exponent first. (−2)−3=1(−2)3=1−8(-2)^{-3} = \frac{1}{(-2)^3} = \frac{1}{-8}. Now, multiply by the coefficient 2: y=2⋅(1−8)=2−8=−14y = 2 \cdot \left(\frac{1}{-8}\right) = \frac{2}{-8} = -\frac{1}{4}.
  • A is incorrect. This could result from incorrectly calculating 2(−2)−32(-2)^{-3} as 2(8)2(8) or −2(8)-2(8).
  • C is incorrect. This is a sign error, likely from evaluating (−2)3(-2)^3 as 8 instead of -8.
  • D is incorrect. This may result from multiplying the base by the exponent, as in 2(−2⋅−3)=2(6)=122(-2 \cdot -3) = 2(6) = 12.

Question 18

If x≠0x \neq 0, what is the value of (5x3)2⋅(5x6)−1(5x^3)^2 \cdot (5x^6)^{-1}?

  1. 0
  2. 1
  3. 5 (correct answer)
  4. 25
Explanation: First, apply the exponents to each factor. The first term becomes 52(x3)2=25x65^2(x^3)^2 = 25x^6. The second term becomes 5−1(x6)−1=15x−65^{-1}(x^6)^{-1} = \frac{1}{5}x^{-6}. Now multiply the results: (25x6)⋅(15x−6)(25x^6) \cdot (\frac{1}{5}x^{-6}). Group the coefficients and variables: (25⋅15)⋅(x6⋅x−6)(25 \cdot \frac{1}{5}) \cdot (x^6 \cdot x^{-6}). The coefficients multiply to 255=5\frac{25}{5}=5. The variables multiply to x6+(−6)=x0x^{6+(-6)} = x^0. Since x≠0x \neq 0, x0=1x^0=1. The final result is 5⋅1=55 \cdot 1 = 5.
  • A is incorrect. This may result from the misconception that x0=0x^0=0.
  • B is incorrect. This results from an error with the coefficients, such as 5/5=15/5=1 instead of 25/525/5.
  • D is incorrect. This results from ignoring the second coefficient, 5−15^{-1}, or making a sign error with the exponents, leading to 25x1225x^{12}, which doesn't simplify to a constant.

Question 19

What is the value of the expression −24(−4)2\frac{-2^4}{(-4)^2}?

  1. -1 (correct answer)
  2. −12-\frac{1}{2}
  3. 12\frac{1}{2}
  4. 1
Explanation: Evaluate the numerator and the denominator separately. In the numerator, −24-2^4, the exponent applies only to the 2, not the negative sign. So, −24=−(2⋅2⋅2⋅2)=−16-2^4 = -(2 \cdot 2 \cdot 2 \cdot 2) = -16. In the denominator, (−4)2(-4)^2, the exponent applies to −4-4. So, (−4)2=(−4)(−4)=16(-4)^2 = (-4)(-4) = 16. The fraction is −1616\frac{-16}{16}, which simplifies to -1.
  • B is incorrect. This could result from an error in calculating the numerator, such as −24=−8-2^4 = -8.
  • C is incorrect. This could result from sign errors in both the numerator and denominator.
  • D is incorrect. This would result from an error in the numerator, such as evaluating −24-2^4 as 16, or an error in the denominator, such as (−4)2=−16(-4)^2 = -16, leading to −16−16=1\frac{-16}{-16} = 1.

Question 20

What is the value of (−1)100−(−1)101(-1)^{100} - (-1)^{101}?

  1. -2
  2. 0
  3. 1
  4. 2 (correct answer)
Explanation: The value of −1-1 raised to a power depends on whether the exponent is even or odd. For an even exponent, the result is 1. For an odd exponent, the result is -1. In this expression, (−1)100=1(-1)^{100} = 1 because 100 is even. And (−1)101=−1(-1)^{101} = -1 because 101 is odd. Substituting these values into the expression gives 1−(−1)1 - (-1), which simplifies to 1+1=21 + 1 = 2.
  • A is incorrect. This would result from reversing the signs, calculating −1−1=−2-1 - 1 = -2.
  • B is incorrect. This results from either thinking both terms are 1 (1−1=01-1=0) or both terms are -1 (−1−(−1)=0-1 - (-1)=0).
  • C is incorrect. This could happen if the subtraction is ignored or if the second term is mistakenly evaluated to 0.