ACCUPLACER Quantitative Reasoning, Algebra & Statistics Quiz: Negative Exponents As Reciprocals
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Negative Exponents As ReciprocalsQuestion 1 of 20

What is the value of (23)−2(\frac{2}{3})^{-2}?

−94-\frac{9}{4}
−49-\frac{4}{9}
49\frac{4}{9}
94\frac{9}{4}
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ACCUPLACER Quantitative Reasoning, Algebra & Statistics Quiz

ACCUPLACER Quantitative Reasoning, Algebra & Statistics Quiz: Negative Exponents As Reciprocals

Practice Negative Exponents As Reciprocals 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 Negative Exponents As Reciprocals, 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

What is the value of (23)−2(\frac{2}{3})^{-2}?

  1. −94-\frac{9}{4}
  2. −49-\frac{4}{9}
  3. 49\frac{4}{9}
  4. 94\frac{9}{4} (correct answer)
Explanation: A negative exponent indicates the reciprocal of the base raised to the positive exponent. So, (23)−2=(32)2(\frac{2}{3})^{-2} = (\frac{3}{2})^2. Squaring the fraction gives 3222=94\frac{3^2}{2^2} = \frac{9}{4}.

Question 2

What is the value of the expression 7−1+707^{-1} + 7^0?

  1. 87\frac{8}{7} (correct answer)
  2. 17\frac{1}{7}
  3. 00
  4. 88
Explanation: First, evaluate each term. By the rule of negative exponents, 7−1=177^{-1} = \frac{1}{7}. By the zero exponent rule, any non-zero number raised to the power of 0 is 1, so 70=17^0 = 1. The expression is now 17+1\frac{1}{7} + 1. To add these, find a common denominator: 17+77=87\frac{1}{7} + \frac{7}{7} = \frac{8}{7}.

Question 3

Which expression is equivalent to y−4y−7\frac{y^{-4}}{y^{-7}}?

  1. y−11y^{-11}
  2. 1y3\frac{1}{y^3}
  3. y3y^3 (correct answer)
  4. 1y11\frac{1}{y^{11}}
Explanation: According to the quotient rule for exponents, when dividing terms with the same base, you subtract the exponent of the denominator from the exponent of the numerator. So, y−4y−7=y−4−(−7)=y−4+7=y3\frac{y^{-4}}{y^{-7}} = y^{-4 - (-7)} = y^{-4+7} = y^3.

Question 4

What is the value of 3−1+3−23^{-1} + 3^{-2}?

  1. 127\frac{1}{27}
  2. 29\frac{2}{9}
  3. 49\frac{4}{9} (correct answer)
  4. 112\frac{1}{12}
Explanation: First, evaluate each term separately. 3−1=133^{-1} = \frac{1}{3} and 3−2=132=193^{-2} = \frac{1}{3^2} = \frac{1}{9}. Then, add the two fractions: 13+19\frac{1}{3} + \frac{1}{9}. To add them, find a common denominator, which is 9. 39+19=49\frac{3}{9} + \frac{1}{9} = \frac{4}{9}.

Question 5

Simplify: (x−2y3x3y−1)−2\left(\frac{x^{-2}y^3}{x^3y^{-1}}\right)^{-2}

  1. y10x8\frac{y^{10}}{x^8}
  2. y8x10\frac{y^8}{x^{10}}
  3. x8y10\frac{x^8}{y^{10}}
  4. x10y8\frac{x^{10}}{y^8} (correct answer)
Explanation: When you see complex expressions with negative exponents and parentheses raised to powers, break the problem into systematic steps using exponent rules. First, simplify the expression inside the parentheses by applying the quotient rule: subtract exponents when dividing like bases. For the x terms: x−2÷x3=x−2−3=x−5x^{-2} ÷ x^3 = x^{-2-3} = x^{-5}. For the y terms: y3÷y−1=y3−(−1)=y4y^3 ÷ y^{-1} = y^{3-(-1)} = y^4. This gives us (x−5y41)−2=(x−5y4)−2\left(\frac{x^{-5}y^4}{1}\right)^{-2} = (x^{-5}y^4)^{-2}. Next, apply the power rule by multiplying each exponent by the outer exponent of -2: x(−5)(−2)y(4)(−2)=x10y−8x^{(-5)(-2)}y^{(4)(-2)} = x^{10}y^{-8}. Since y−8=1y8y^{-8} = \frac{1}{y^8}, the final answer is x10y8\frac{x^{10}}{y^8}, which is choice D. Choice A (y10x8\frac{y^{10}}{x^8}) results from incorrectly adding exponents instead of subtracting in the quotient rule. Choice B (y8x10\frac{y^8}{x^{10}}) comes from forgetting to apply the negative sign when using the power rule. Choice C (x8y10\frac{x^8}{y^{10}}) combines both errors: incorrect quotient rule application and sign mistakes in the power rule. Remember this pattern: when you have nested exponents with fractions, always simplify the inside first using quotient rules, then apply the outer exponent to everything. Double-check your signs at each step, as negative exponents are common trap areas on the Accuplacer.

Question 6

If m−3⋅n−2=172m^{-3} \cdot n^{-2} = \frac{1}{72} and m=2m = 2, what is the value of n2n^2?

  1. 33
  2. 99 (correct answer)
  3. 19\frac{1}{9}
  4. 1818
Explanation: This question tests your understanding of negative exponents and algebraic manipulation. When you see negative exponents combined with an equation to solve, remember that negative exponents create fractions: x−a=1xax^{-a} = \frac{1}{x^a}. Start by rewriting the given equation using positive exponents: m−3⋅n−2=1m3⋅1n2=1m3n2=172m^{-3} \cdot n^{-2} = \frac{1}{m^3} \cdot \frac{1}{n^2} = \frac{1}{m^3 n^2} = \frac{1}{72} Since m=2m = 2, substitute this value: 123n2=172\frac{1}{2^3 n^2} = \frac{1}{72} Calculate 23=82^3 = 8, so: 18n2=172\frac{1}{8n^2} = \frac{1}{72} When two fractions with numerator 1 are equal, their denominators must be equal: 8n2=728n^2 = 72 Solve for n2n^2: n2=728=9n^2 = \frac{72}{8} = 9 The answer is B) 99. Looking at the wrong answers: A) 33 would be the value of nn, not n2n^2 (since 9=3\sqrt{9} = 3). This tests whether you confused what the question was asking for. C) 19\frac{1}{9} results from incorrectly thinking that n−2=19n^{-2} = \frac{1}{9} means n2=19n^2 = \frac{1}{9}, confusing negative and positive exponents. D) 1818 comes from calculation errors, possibly from incorrectly handling the fraction arithmetic. Key strategy: When working with negative exponents, always convert to positive exponents first using x−a=1xax^{-a} = \frac{1}{x^a}. This makes the algebra much clearer and reduces errors.

Question 7

Which expression is equivalent to 2−3⋅4−28−1\frac{2^{-3} \cdot 4^{-2}}{8^{-1}}?

  1. 116\frac{1}{16} (correct answer)
  2. 132\frac{1}{32}
  3. 1616
  4. 18\frac{1}{8}
Explanation: Convert all terms to base 2: 2−3=182^{-3} = \frac{1}{8}, 4−2=(22)−2=2−4=1164^{-2} = (2^2)^{-2} = 2^{-4} = \frac{1}{16}, and 8−1=(23)−1=2−3=188^{-1} = (2^3)^{-1} = 2^{-3} = \frac{1}{8}. The expression becomes 18⋅11618=112818=1128⋅81=8128=116\frac{\frac{1}{8} \cdot \frac{1}{16}}{\frac{1}{8}} = \frac{\frac{1}{128}}{\frac{1}{8}} = \frac{1}{128} \cdot \frac{8}{1} = \frac{8}{128} = \frac{1}{16}. Choice B results from forgetting to multiply by 8 in the final step. Choice C comes from incorrectly treating negative exponents as positive. Choice D results from computational errors in the fraction arithmetic.

Question 8

If 2x=1162^x = \frac{1}{16}, what is the value of xx?

  1. −8-8
  2. −4-4 (correct answer)
  3. 14\frac{1}{4}
  4. 44
Explanation: To solve for xx, we need to express both sides of the equation with the same base. We know that 16=2416 = 2^4. Therefore, 116=124\frac{1}{16} = \frac{1}{2^4}. Using the rule of negative exponents, 124=2−4\frac{1}{2^4} = 2^{-4}. So the equation becomes 2x=2−42^x = 2^{-4}, which means x=−4x = -4.

Question 9

Which of the following expressions is equivalent to (4a)−2(4a)^{-2}?

  1. −16a2-16a^2
  2. 14a2\frac{1}{4a^2}
  3. 116a2\frac{1}{16a^2} (correct answer)
  4. a216\frac{a^2}{16}
Explanation: The exponent −2-2 applies to the entire term inside the parentheses, both the 4 and the aa. So, (4a)−2=4−2a−2(4a)^{-2} = 4^{-2}a^{-2}. Applying the negative exponent rule, this becomes 142⋅1a2=116⋅1a2=116a2\frac{1}{4^2} \cdot \frac{1}{a^2} = \frac{1}{16} \cdot \frac{1}{a^2} = \frac{1}{16a^2}.

Question 10

Which of the following is equivalent to a−1+b−1a−1\frac{a^{-1} + b^{-1}}{a^{-1}}?

  1. bb
  2. a+bb\frac{a+b}{b} (correct answer)
  3. ba+b\frac{b}{a+b}
  4. 1+b−11+b^{-1}
Explanation: First, rewrite the expression with positive exponents: 1a+1b1a\frac{\frac{1}{a} + \frac{1}{b}}{\frac{1}{a}}. Simplify the numerator by finding a common denominator: 1a+1b=bab+aab=a+bab\frac{1}{a} + \frac{1}{b} = \frac{b}{ab} + \frac{a}{ab} = \frac{a+b}{ab}. Now the expression is a+bab1a\frac{\frac{a+b}{ab}}{\frac{1}{a}}. To divide by a fraction, multiply by its reciprocal: a+bab⋅a1\frac{a+b}{ab} \cdot \frac{a}{1}. The aa in the numerator and denominator cancels out, leaving a+bb\frac{a+b}{b}.

Question 11

What is the value of (−4)−2(-4)^{-2}?

  1. −16-16
  2. −116-\frac{1}{16}
  3. 116\frac{1}{16} (correct answer)
  4. 1616
Explanation: The expression (−4)−2(-4)^{-2} means 1(−4)2\frac{1}{(-4)^2}. The base is −4-4, which is squared. (−4)2=(−4)×(−4)=16(-4)^2 = (-4) \times (-4) = 16. Therefore, the value is 116\frac{1}{16}. The result is positive because a negative number squared is positive.

Question 12

Which of the following expressions is NOT equivalent to −19-\frac{1}{9}?

  1. −3−2-3^{-2}
  2. (−3)−2(-3)^{-2} (correct answer)
  3. −(9−1)-(9^{-1})
  4. −(13)2-(\frac{1}{3})^2
Explanation: We need to evaluate each option. A) −3−2=−(132)=−19-3^{-2} = -(\frac{1}{3^2}) = -\frac{1}{9}. C) −(9−1)=−(19)=−19-(9^{-1}) = -(\frac{1}{9}) = -\frac{1}{9}. D) −(13)2=−(19)=−19-(\frac{1}{3})^2 = -(\frac{1}{9}) = -\frac{1}{9}. Option B is (−3)−2=1(−3)2=19(-3)^{-2} = \frac{1}{(-3)^2} = \frac{1}{9}. Since 19\frac{1}{9} is not equal to −19-\frac{1}{9}, this is the correct choice.

Question 13

If k=−2k = -2, what is the value of k−3k^{-3}?

  1. −8-8
  2. −18-\frac{1}{8} (correct answer)
  3. 18\frac{1}{8}
  4. 88
Explanation: Substitute k=−2k = -2 into the expression to get (−2)−3(-2)^{-3}. The negative exponent means to take the reciprocal of the base: 1(−2)3\frac{1}{(-2)^3}. Now, calculate the power in the denominator: (−2)3=(−2)(−2)(−2)=−8(-2)^3 = (-2)(-2)(-2) = -8. The final value is 1−8\frac{1}{-8}, or −18-\frac{1}{8}.

Question 14

The width of a certain particle is 4×10−54 \times 10^{-5} millimeters. Which of the following values is equivalent to this width?

  1. (-400,000) millimeters
  2. 0.000040.00004 millimeters (correct answer)
  3. 0.00040.0004 millimeters
  4. (400,000) millimeters
Explanation: The expression 10−510^{-5} means 1105\frac{1}{10^5}, which is 1100,000\frac{1}{100,000}. So, the width is 4×1100,000=4100,0004 \times \frac{1}{100,000} = \frac{4}{100,000} millimeters. As a decimal, this is 0.000040.00004. This means moving the decimal point in 4.0 five places to the left.

Question 15

For what value of nn is 5×10n=0.0055 \times 10^n = 0.005?

  1. −4-4
  2. −3-3 (correct answer)
  3. −2-2
  4. 33
Explanation: To find the value of nn, first isolate 10n10^n by dividing both sides of the equation by 5: 10n=0.0055=0.00110^n = \frac{0.005}{5} = 0.001. Next, express 0.001 as a power of 10. 0.001=11000=11030.001 = \frac{1}{1000} = \frac{1}{10^3}. Using the negative exponent rule, 1103=10−3\frac{1}{10^3} = 10^{-3}. So, 10n=10−310^n = 10^{-3}, which means n=−3n = -3.

Question 16

Which of the following numbers has the greatest value?

  1. 2−52^{-5}
  2. 5−25^{-2}
  3. 3−33^{-3}
  4. 4−24^{-2} (correct answer)
Explanation: To compare these numbers, evaluate each one: 2−5=125=1322^{-5} = \frac{1}{2^5} = \frac{1}{32}; 5−2=152=1255^{-2} = \frac{1}{5^2} = \frac{1}{25}; 3−3=133=1273^{-3} = \frac{1}{3^3} = \frac{1}{27}; 4−2=142=1164^{-2} = \frac{1}{4^2} = \frac{1}{16}. When comparing positive fractions with the same numerator (in this case, 1), the fraction with the smallest denominator has the greatest value. Since 16 is the smallest denominator, 116\frac{1}{16} is the greatest value.

Question 17

If xx is a positive integer greater than 1, which of the following expressions has the least value?

  1. x−1x^{-1}
  2. x−2x^{-2}
  3. (−x)−2(-x)^{-2}
  4. −x−2-x^{-2} (correct answer)
Explanation: Let's analyze the sign of each expression. Since xx is positive, x−1=1xx^{-1} = \frac{1}{x} is positive. Similarly, x−2=1x2x^{-2} = \frac{1}{x^2} is positive. For (−x)−2=1(−x)2=1x2(-x)^{-2} = \frac{1}{(-x)^2} = \frac{1}{x^2}, the result is also positive. However, −x−2=−(1x2)-x^{-2} = -(\frac{1}{x^2}) is negative. Since a negative number is always less than a positive number, −x−2-x^{-2} has the least value.

Question 18

Which expression is equivalent to 2x4y−26x−1y3\frac{2x^4 y^{-2}}{6x^{-1} y^3}?

  1. x3y3\frac{x^3 y}{3}
  2. y53x5\frac{y^5}{3x^5}
  3. x53y5\frac{x^5}{3y^5} (correct answer)
  4. 3x3y3x^3y
Explanation: Simplify the expression in three parts: coefficients, xx terms, and yy terms. Coefficients: 26=13\frac{2}{6} = \frac{1}{3}. For xx: x4x−1=x4−(−1)=x5\frac{x^4}{x^{-1}} = x^{4 - (-1)} = x^5. For yy: y−2y3=y−2−3=y−5\frac{y^{-2}}{y^3} = y^{-2 - 3} = y^{-5}. Combining these gives 13x5y−5\frac{1}{3}x^5y^{-5}. To write with a positive exponent, move y−5y^{-5} to the denominator: x53y5\frac{x^5}{3y^5}.

Question 19

Which of the following expressions is equivalent to 5x−35x^{-3}?

  1. −15x-15x
  2. −5x3-5x^3
  3. 15x3\frac{1}{5x^3}
  4. 5x3\frac{5}{x^3} (correct answer)
Explanation: The negative exponent rule states that a−n=1ana^{-n} = \frac{1}{a^n}. In the expression 5x−35x^{-3}, the exponent −3-3 applies only to xx, not to the coefficient 5. Therefore, x−3=1x3x^{-3} = \frac{1}{x^3}, and the entire expression is 5⋅1x3=5x35 \cdot \frac{1}{x^3} = \frac{5}{x^3}.

Question 20

If f(x)=x−2f(x) = x^{-2}, which of the following is equal to f(1)−f(2)f(1) - f(2)?

  1. 34\frac{3}{4} (correct answer)
  2. −34-\frac{3}{4}
  3. −3-3
  4. 11
Explanation: First, we need to evaluate the function for each input. f(1)=1−2=112=1f(1) = 1^{-2} = \frac{1}{1^2} = 1. Next, f(2)=2−2=122=14f(2) = 2^{-2} = \frac{1}{2^2} = \frac{1}{4}. The expression f(1)−f(2)f(1) - f(2) becomes 1−141 - \frac{1}{4}. To subtract, we can write 1 as 44\frac{4}{4}. So, 44−14=34\frac{4}{4} - \frac{1}{4} = \frac{3}{4}.