Middle School Math Quiz: Apply Properties Of Integer Exponents
20 questions · exam conditions
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Apply Properties Of Integer ExponentsQuestion 1 of 20

Which expression is equivalent to 2−3⋅272−1⋅22\frac{2^{-3} \cdot 2^7}{2^{-1} \cdot 2^2}?

232^3
2−12^{-1}
2112^{11}
1211\frac{1}{2^{11}}
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Middle School Math Quiz

Middle School Math Quiz: Apply Properties Of Integer Exponents

Practice Apply Properties Of Integer Exponents in Middle School Math 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 Apply Properties Of Integer Exponents, giving you a quick way to practice the rules, question types, and explanations that matter most for Middle School Math.

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 expression is equivalent to 2−3⋅272−1⋅22\frac{2^{-3} \cdot 2^7}{2^{-1} \cdot 2^2}?

  1. 232^3 (correct answer)
  2. 2−12^{-1}
  3. 2112^{11}
  4. 1211\frac{1}{2^{11}}
Explanation: Using properties of exponents: 2−3⋅272−1⋅22=2−3+72−1+2=2421=24−1=23\frac{2^{-3} \cdot 2^7}{2^{-1} \cdot 2^2} = \frac{2^{-3+7}}{2^{-1+2}} = \frac{2^4}{2^1} = 2^{4-1} = 2^3. Choice B results from incorrectly computing the numerator as 2−3−72^{-3-7}. Choice C comes from adding all exponents instead of applying quotient rules. Choice D results from computing 2−112^{-11} by incorrectly subtracting exponents.

Question 2

If x=3−2x = 3^{-2} and y=3−4y = 3^{-4}, what is the value of x2y\frac{x^2}{y} expressed as a power of 3?

  1. 303^{0} (correct answer)
  2. 3−83^{-8}
  3. 343^{4}
  4. 3123^{12}
Explanation: Substituting: x2y=(3−2)23−4=3−43−4=3−4−(−4)=30=1\frac{x^2}{y} = \frac{(3^{-2})^2}{3^{-4}} = \frac{3^{-4}}{3^{-4}} = 3^{-4-(-4)} = 3^0 = 1. Choice B results from multiplying exponents instead of using the power rule correctly. Choice C comes from computing 3−4+43^{-4+4} but forgetting the power of 2. Choice D results from computing 3(−2)2−(−4)=34+43^{(-2)^2 - (-4)} = 3^{4+4}, incorrectly applying the power rule.

Question 3

Simplify: (3−2)4⋅3−13−5⋅3−4\frac{(3^{-2})^4 \cdot 3^{-1}}{3^{-5} \cdot 3^{-4}}

  1. 3−83^{-8}
  2. 3−183^{-18}
  3. 13\frac{1}{3}
  4. 303^{0} (correct answer)
Explanation: When you encounter expressions with exponents that need simplifying, your goal is to use the rules of exponents to combine like terms. This problem tests your understanding of the power rule, product rule, and quotient rule for exponents. Let's work through this step-by-step. First, apply the power rule to (3−2)4(3^{-2})^4. When raising a power to a power, you multiply the exponents: (3−2)4=3−2⋅4=3−8(3^{-2})^4 = 3^{-2 \cdot 4} = 3^{-8}. Now your expression becomes: 3−8⋅3−13−5⋅3−4\frac{3^{-8} \cdot 3^{-1}}{3^{-5} \cdot 3^{-4}} Next, use the product rule in both the numerator and denominator. When multiplying powers with the same base, you add the exponents:
  • Numerator: 3−8⋅3−1=3−8+(−1)=3−93^{-8} \cdot 3^{-1} = 3^{-8 + (-1)} = 3^{-9}
  • Denominator: 3−5⋅3−4=3−5+(−4)=3−93^{-5} \cdot 3^{-4} = 3^{-5 + (-4)} = 3^{-9}
This gives you: 3−93−9=3−9−(−9)=3−9+9=30=1\frac{3^{-9}}{3^{-9}} = 3^{-9 - (-9)} = 3^{-9 + 9} = 3^0 = 1 Answer A (3−83^{-8}) comes from stopping after the first step and forgetting to continue simplifying. Answer B (3−183^{-18}) results from incorrectly multiplying all the exponents together instead of using the proper rules. Answer C (13\frac{1}{3}) might come from calculation errors when combining exponents. Remember: when working with exponent problems, apply the rules systematically—power rule first, then product/quotient rules. Double-check your arithmetic when adding and subtracting negative exponents, as sign errors are common here.

Question 4

Simplify the expression: (4−2)3\left(4^{-2}\right)^3.

  1. 14−6\dfrac{1}{4^{-6}}
  2. 464^{6}
  3. 4−64^{-6} (correct answer)
  4. 4−54^{-5}
Explanation: This tests applying integer exponent properties: product rule (am×an=am+na^m \times a^n = a^{m+n}), quotient rule (am÷an=am−na^m \div a^n = a^{m-n}), power rule ((am)n=amn(a^m)^n = a^{m n}), zero exponent (a0=1a^0 = 1), and negative exponents (a−n=1ana^{-n} = \frac{1}{a^n}). For expression 34×3−23^4 \times 3^{-2}, apply product rule: same base 3, add exponents 4 + (-2) = 2, giving 323^2, then evaluate: 32=93^2 = 9. For (23)4(2^3)^4, multiply exponents: 3 \times 4 = 12 giving 2122^{12}, evaluating: 212=40962^{12} = 4096. Key principle: properties apply when bases are identical; different bases (23×322^3 \times 3^2) can't combine using these rules. For ((4−2)3)(\left(4^{-2}\right)^3), apply the power rule: multiply exponents -2 \times 3 = -6, giving 4−64^{-6}. Choice A is correct because it properly applies the power rule by multiplying exponents, including the negative one. Choice D ignores negative making 4−6=464^{-6} = 4^6 when should be 1/461/4^6. Strategy: (1) identify operation (multiply/divide/power of power), (2) check bases match (must be same for properties), (3) apply correct rule (add/subtract/multiply exponents), (4) simplify further (combine like terms, evaluate if needed), (5) handle special cases (zero→1, negative→reciprocal). Common mistakes: multiplying bases when should add exponents, confusing rules (adding when should multiply for powers), working with different bases (23×322^3 \times 3^2 can't simplify using exponent rules).

Question 5

A student writes an expression for a pattern: (32×3−5)÷3−1\left(3^2 \times 3^{-5}\right) \div 3^{-1}. Simplify it and write your answer as a single power of 3.

  1. 1−32\dfrac{1}{-3^2}
  2. 323^{2} (correct answer)
  3. 3−23^{-2}
  4. 3−63^{-6}
Explanation: This tests applying integer exponent properties: product rule (am×an=am+na^m \times a^n = a^{m+n}), quotient rule (am÷an=am−na^m \div a^n = a^{m-n}), power rule ((am)n=amn(a^m)^n = a^{m n}), zero exponent (a0=1a^0 = 1), and negative exponents (a−n=1ana^{-n} = \frac{1}{a^n}). For expression 34×3−23^4 \times 3^{-2}, apply product rule: same base 3, add exponents 4+(−2)=24 + (-2) = 2, giving 323^2, then evaluate: 32=93^2 = 9. For (23)4(2^3)^4, multiply exponents: 3×4=123 \times 4 = 12 giving 2122^{12}, evaluating: 212=40962^{12} = 4096. Key principle: properties apply when bases are identical; different bases (23×322^3 \times 3^2) can't combine using these rules. For ((32×3−5)÷3−1)(\left(3^2 \times 3^{-5}\right) \div 3^{-1}), first apply the product rule inside the parentheses: 2+(−5)=−32 + (-5) = -3, giving (3−3)(3^{-3}), then use the quotient rule: −3−(−1)=−2-3 - (-1) = -2, resulting in (3−2)(3^{-2}). Choice B is correct because it properly applies the product rule by adding exponents and then the quotient rule by subtracting exponents, correctly handling the negative exponent. Choice A ignores negative making 3−3=333^{-3} = 3^3 when should be 1/271/27 or confuses subtraction in quotient. Strategy: (1) identify operation (multiply/divide/power of power), (2) check bases match (must be same for properties), (3) apply correct rule (add/subtract/multiply exponents), (4) simplify further (combine like terms, evaluate if needed), (5) handle special cases (zero → 1, negative → reciprocal). Common mistakes: multiplying bases when should add exponents, confusing rules (adding when should multiply for powers), working with different bases (23×322^3 \times 3^2 can't simplify using exponent rules).

Question 6

Simplify the expression and write your answer as a single power: (32×3−5)÷3−1(3^2 \times 3^{-5}) \div 3^{-1}

  1. 343^{4}
  2. 13\dfrac{1}{3}
  3. 3−23^{-2} (correct answer)
  4. 3−73^{-7}
Explanation: This tests applying integer exponent properties: product rule (am×an=am+na^m \times a^n = a^{m+n}), quotient rule (am÷an=am−na^m \div a^n = a^{m-n}), power rule ((am)n=amna^m)^n = a^{m n}), zero exponent (a0=1a^0 = 1), and negative exponents (a−n=1/ana^{-n} = 1/a^n). For expression (32×3−53^2 \times 3^{-5}) \div 3−13^{-1}, first simplify the parentheses using product rule: 32×3−5=32+(−5)=3−33^2 \times 3^{-5} = 3^{2 + (-5)} = 3^{-3}. Then apply quotient rule: 3−3÷3−1=3−3−(−1)=3−3+1=3−23^{-3} \div 3^{-1} = 3^{-3 - (-1)} = 3^{-3 + 1} = 3^{-2}. Choice B (3−23^{-2}) is correct because it properly applies the product rule first (adding 2 + (-5) = -3), then the quotient rule (subtracting -3 - (-1) = -2). Choice A (3−73^{-7}) incorrectly adds all exponents without considering division, choice C (343^4) has wrong sign, and choice D (13\dfrac{1}{3}) equals 3−13^{-1} not 3−23^{-2}. Strategy: (1) identify operation (multiply/divide/power of power), (2) check bases match (must be same for properties), (3) apply correct rule (add/subtract/multiply exponents), (4) simplify further (combine like terms, evaluate if needed), (5) handle special cases (zero → 1, negative → reciprocal). Common mistakes: multiplying bases when should add exponents, confusing rules (adding when should multiply for powers), working with different bases (23×322^3 \times 3^2 can't simplify using exponent rules).

Question 7

Evaluate the expression: 4−34^{-3}

  1. 6464
  2. 164\dfrac{1}{64} (correct answer)
  3. 112\dfrac{1}{12}
  4. −164-\dfrac{1}{64}
Explanation: This tests applying integer exponent properties: product rule (am×an=am+na^m \times a^n = a^{m+n}), quotient rule (am÷an=am−na^m \div a^n = a^{m-n}), power rule ((am)n=amna^m)^n = a^{m n}), zero exponent (a0=1a^0 = 1), and negative exponents (a−n=1ana^{-n} = \frac{1}{a^n}). For expression 4−34^{-3}, apply negative exponent rule: 4−3=143=1644^{-3} = \frac{1}{4^3} = \frac{1}{64}. The calculation shows: 43=4×4×4=644^3 = 4 \times 4 \times 4 = 64, so 4−3=1644^{-3} = \frac{1}{64}. Choice A (164\frac{1}{64}) is correct because it properly applies the negative exponent rule (a−n=1ana^{-n} = \frac{1}{a^n}) and correctly evaluates 43=644^3 = 64. Choice B (6464) ignores the negative sign making 4−3=434^{-3} = 4^3, choice C (−164-\frac{1}{64}) incorrectly makes the result negative when negative exponent means reciprocal not negative value, and choice D (112\frac{1}{12}) equals 1/(4×3)1/(4\times3) instead of 1/431/4^3. Strategy: (1) identify operation (multiply/divide/power of power), (2) check bases match (must be same for properties), (3) apply correct rule (add/subtract/multiply exponents), (4) simplify further (combine like terms, evaluate if needed), (5) handle special cases (zero→1, negative→reciprocal). Common mistakes: multiplying bases when should add exponents, confusing rules (adding when should multiply for powers), working with different bases (23×322^3 \times 3^2 can't simplify using exponent rules).

Question 8

Simplify the expression and write your answer as a single power: (60×6−2)×65\left(6^0 \times 6^{-2}\right) \times 6^5

  1. 606^{0}
  2. 636^{3} (correct answer)
  3. 6−76^{-7}
  4. 676^{7}
Explanation: This tests applying integer exponent properties: product rule (am×an=am+na^m \times a^n = a^{m+n}), quotient rule (am÷an=am−na^m \div a^n = a^{m-n}), power rule ((am)n=amn(a^m)^n = a^{m n}), zero exponent (a0=1a^0 = 1), and negative exponents (a−n=1ana^{-n} = \frac{1}{a^n}). For expression (60×6−26^0 \times 6^{-2}) ×65\times 6^5, first simplify parentheses: 60=16^0 = 1, so 60×6−2=60+(−2)=6−26^0 \times 6^{-2} = 6^{0 + (-2)} = 6^{-2}. Then multiply by 656^5: 6−2×65=6−2+5=636^{-2} \times 6^5 = 6^{-2 + 5} = 6^3. Choice A (636^3) is correct because it properly applies the product rule throughout (0+(−2)=−20 + (-2) = -2, then −2+5=3-2 + 5 = 3). Choice B (6−76^{-7}) incorrectly subtracts all exponents, choice C (676^7) has wrong calculation, and choice D (606^0) would mean the result equals 1. Strategy: (1) identify operation (multiply/divide/power of power), (2) check bases match (must be same for properties), (3) apply correct rule (add/subtract/multiply exponents), (4) simplify further (combine like terms, evaluate if needed), (5) handle special cases (zero → 1, negative → reciprocal). Common mistakes: multiplying bases when should add exponents, confusing rules (adding when should multiply for powers), working with different bases (23×322^3 \times 3^2 can't simplify using exponent rules).

Question 9

Which expression is equivalent to (23)4÷210(2^3)^4 \div 2^{10}, written as a single power?

  1. 2222^{22}
  2. 272^7
  3. 2122^{12}
  4. 222^2 (correct answer)
Explanation: Using the power rule, (23)4=23×4=212(2^3)^4 = 2^{3\times4} = 2^{12}. Using the quotient rule, 212÷210=212−10=222^{12}\div2^{10}=2^{12-10}=2^2. So (23)4÷210=22(2^3)^4\div2^{10}=2^2. Choice B (272^7) comes from adding 3+43+4 to get 7 instead of multiplying. Choice A (2222^{22}) and Choice C (2122^{12}) each reflect skipping or mishandling one of the two steps needed to fully simplify the expression.

Question 10

A scientist uses the expression 5−2⋅5n⋅5−35^{-2} \cdot 5^n \cdot 5^{-3} to represent a measurement. For what value of nn will this expression equal 525^2?

  1. n=2n = 2
  2. n=7n = 7 (correct answer)
  3. n=3n = 3
  4. n=−3n = -3
Explanation: Using the product rule: 5−2⋅5n⋅5−3=5−2+n+(−3)=5n−55^{-2} \cdot 5^n \cdot 5^{-3} = 5^{-2+n+(-3)} = 5^{n-5}. For this to equal 525^2, we need n−5=2n-5 = 2, so n=7n = 7. Choice A gives 5−35^{-3}, choice C gives 5−25^{-2}, and choice D gives 5−85^{-8}.

Question 11

Which expression represents the reciprocal of (2−3)−2(2^{-3})^{-2}?

  1. 2−62^{-6} (correct answer)
  2. 262^{6}
  3. 126\frac{1}{2^{6}}
  4. 12−6\frac{1}{2^{-6}}
Explanation: First, (2−3)−2=2(−3)(−2)=26(2^{-3})^{-2} = 2^{(-3)(-2)} = 2^6. The reciprocal of 262^6 is 126=2−6\frac{1}{2^6} = 2^{-6}. Choice B is the original expression, not its reciprocal. Choice C equals 2−62^{-6} but isn't simplified. Choice D equals 262^6, which is the original expression.

Question 12

What is the value of 6−2⋅6−46−8\frac{6^{-2} \cdot 6^{-4}}{6^{-8}} written in simplest form?

  1. 6−146^{-14}
  2. 626^{2} (correct answer)
  3. 6106^{10}
  4. 6−26^{-2}
Explanation: 6−2⋅6−46−8=6−66−8=6−6−(−8)=62\frac{6^{-2} \cdot 6^{-4}}{6^{-8}} = \frac{6^{-6}}{6^{-8}} = 6^{-6-(-8)} = 6^{2}. Choice A results from multiplying all exponents. Choice C results from adding all exponents. Choice D represents 6−26^{-2}, which would result from computing 6−6+(−8)6^{-6+(-8)}.

Question 13

Which statement about the expression 4−2⋅4x4−3\frac{4^{-2} \cdot 4^{x}}{4^{-3}} is true when x=−1x = -1?

  1. The expression equals 424^{2} and has a value of 16
  2. The expression equals 4−64^{-6} and has a value of 14096\frac{1}{4096}
  3. The expression equals 404^{0} and has a value of 1 (correct answer)
  4. The expression equals 4−24^{-2} and has a value of 116\frac{1}{16}
Explanation: When you encounter expressions with exponents that need to be simplified, your goal is to use the rules of exponents to combine terms and then substitute the given value. Let's work through this step-by-step using the exponent rules. First, simplify 4−2⋅4x4−3\frac{4^{-2} \cdot 4^{x}}{4^{-3}} before substituting x=−1x = -1. Using the product rule, 4−2⋅4x=4−2+x4^{-2} \cdot 4^{x} = 4^{-2+x}. Then using the quotient rule, 4−2+x4−3=4(−2+x)−(−3)=4−2+x+3=41+x\frac{4^{-2+x}}{4^{-3}} = 4^{(-2+x)-(-3)} = 4^{-2+x+3} = 4^{1+x}. Now substitute x=−1x = -1: 41+(−1)=40=14^{1+(-1)} = 4^{0} = 1. Since any non-zero number raised to the zero power equals 1, the expression equals 404^{0} with a value of 1. Looking at the wrong answers: Answer A incorrectly calculates the exponent as 2, perhaps by misapplying the quotient rule or making sign errors. Answer B gets 4−64^{-6}, which might result from adding all the exponents incorrectly (−2+(−1)+(−3)=−6-2 + (-1) + (-3) = -6) instead of properly applying the quotient rule. Answer D gives 4−24^{-2}, possibly from incorrectly combining the original exponents or stopping the calculation too early. The key strategy here is to always simplify expressions with exponents algebraically first, then substitute the variable's value. This approach prevents calculation errors and makes the work cleaner. Remember: when dividing powers with the same base, subtract the exponents, and when multiplying, add them.

Question 14

If a−3=18a^{-3} = \frac{1}{8}, what is the value of a−6a^{-6}?

  1. 116\frac{1}{16}
  2. 164\frac{1}{64} (correct answer)
  3. 1512\frac{1}{512}
  4. 14096\frac{1}{4096}
Explanation: Since a−3=18a^{-3} = \frac{1}{8}, we have a3=8a^3 = 8, so a=2a = 2. Therefore, a−6=2−6=126=164a^{-6} = 2^{-6} = \frac{1}{2^6} = \frac{1}{64}. Alternatively, a−6=(a−3)2=(18)2=164a^{-6} = (a^{-3})^2 = (\frac{1}{8})^2 = \frac{1}{64}. Choice A represents a−4a^{-4}, choice C represents a−9a^{-9}, and choice D represents a−12a^{-12}.

Question 15

A science club records a value using exponents: (92)0×9−4\left(9^2\right)^0 \times 9^{-4}. Simplify and write your answer as a single power of 9.

  1. 00
  2. 9−49^{-4} (correct answer)
  3. 909^{0}
  4. 949^{4}
Explanation: This tests applying integer exponent properties: product rule (aᵐ × aⁿ = aᵐ⁺ⁿ), quotient rule (aᵐ ÷ aⁿ = aᵐ⁻ⁿ), power rule ((aᵐ)ⁿ = aᵐⁿ), zero exponent (a⁰=1), and negative exponents (a⁻ⁿ=1/aⁿ). For expression 3⁴ × 3⁻², apply product rule: same base 3, add exponents 4+(-2)=2, giving 3², then evaluate: 3²=9. For (2³)⁴, multiply exponents: 3×4=12 giving 2¹², evaluating: 2¹²=4096. Key principle: properties apply when bases are identical; different bases (2³ × 3²) can't combine using these rules. For (\left(9^2\right)^0 \times 9^{-4}), first apply the power rule and zero exponent: (9^2)^0 = 1, then multiply by 9^{-4}, giving 9^{-4}. Choice C is correct because it properly applies the power rule to get zero exponent equaling 1 and then the product rule effectively. Choice A treats the whole as 9^0 when the zero is only on the power of power. Strategy: (1) identify operation (multiply/divide/power of power), (2) check bases match (must be same for properties), (3) apply correct rule (add/subtract/multiply exponents), (4) simplify further (combine like terms, evaluate if needed), (5) handle special cases (zero→1, negative→reciprocal). Common mistakes: multiplying bases when should add exponents, confusing rules (adding when should multiply for powers), working with different bases (2³ × 3² can't simplify using exponent rules).

Question 16

Simplify the expression and write your answer as a single power: (60×82)÷8−1\left(6^0 \times 8^2\right) \div 8^{-1}.

  1. 838^{3} (correct answer)
  2. 818^{1}
  3. 8−38^{-3}
  4. 48348^{3}
Explanation: This tests applying integer exponent properties: product rule (aᵐ × aⁿ = aᵐ⁺ⁿ), quotient rule (aᵐ ÷ aⁿ = aᵐ⁻ⁿ), power rule ((aᵐ)ⁿ = aᵐⁿ), zero exponent (a⁰=1), and negative exponents (a⁻ⁿ=1/aⁿ). For expression 3⁴ × 3⁻², apply product rule: same base 3, add exponents 4+(-2)=2, giving 3², then evaluate: 3²=9. For (2³)⁴, multiply exponents: 3×4=12 giving 2¹², evaluating: 2¹²=4096. Key principle: properties apply when bases are identical; different bases (2³ × 3²) can't combine using these rules. For (\left(606^0 ×\times 828^2\right) \div 8^{-1}), first evaluate (606^0 = 1), so 1 × 8^2 = 8^2, then apply quotient rule: 2 - (-1) = 3, giving (838^3). Choice C is correct because it correctly evaluates a⁰=1, applies the product rule implicitly, and then the quotient rule by subtracting exponents. Choice D multiplies bases instead of adding exponents (6×8=48, wrong). Strategy: (1) identify operation (multiply/divide/power of power), (2) check bases match (must be same for properties), (3) apply correct rule (add/subtract/multiply exponents), (4) simplify further (combine like terms, evaluate if needed), (5) handle special cases (zero→1, negative→reciprocal). Common mistakes: multiplying bases when should add exponents, confusing rules (adding when should multiply for powers), working with different bases (2³ × 3² can't simplify using exponent rules).

Question 17

Evaluate the expression: 70×257^0 \times 2^5.

  1. 16
  2. 0
  3. 224
  4. 32 (correct answer)
Explanation: This tests applying integer exponent properties: product rule (aᵐ × aⁿ = aᵐ⁺ⁿ), quotient rule (aᵐ ÷ aⁿ = aᵐ⁻ⁿ), power rule ((aᵐ)ⁿ = aᵐⁿ), zero exponent (a⁰=1), and negative exponents (a⁻ⁿ=1/aⁿ). For expression 3⁴ × 3⁻², apply product rule: same base 3, add exponents 4+(-2)=2, giving 3², then evaluate: 3²=9. For (2³)⁴, multiply exponents: 3×4=12 giving 2¹², evaluating: 2¹²=4096. Key principle: properties apply when bases are identical; different bases (2³ × 3²) can't combine using these rules. For (707^0 ×\times 252^5), first recognize the zero exponent gives 1, then compute (252^5 = 32), so 1 × 32 = 32. Choice C is correct because it correctly evaluates a⁰=1 and then multiplies by the evaluated power. Choice A treats a⁰ as 0 when equals 1. Strategy: (1) identify operation (multiply/divide/power of power), (2) check bases match (must be same for properties), (3) apply correct rule (add/subtract/multiply exponents), (4) simplify further (combine like terms, evaluate if needed), (5) handle special cases (zero→1, negative→reciprocal). Common mistakes: multiplying bases when should add exponents, confusing rules (adding when should multiply for powers), working with different bases (2³ × 3² can't simplify using exponent rules).

Question 18

Simplify the expression and write your answer as a single power: (82)385\frac{\left(8^2\right)^3}{8^5}

  1. 818^{1} (correct answer)
  2. 8−18^{-1}
  3. 8118^{11}
  4. 868^{6}
Explanation: This tests applying integer exponent properties: product rule (am×an=am+na^m \times a^n = a^{m+n}), quotient rule (am÷an=am−na^m \div a^n = a^{m-n}), power rule ((am)n=amn(a^m)^n = a^{m n}), zero exponent (a0=1a^0 = 1), and negative exponents (a−n=1ana^{-n} = \frac{1}{a^n}). For expression (82)3/85(8^2)^3 / 8^5, first apply power rule: (82)3=82×3=86(8^2)^3 = 8^{2 \times 3} = 8^6. Then apply quotient rule: 86÷85=86−5=818^6 \div 8^5 = 8^{6-5} = 8^1. Choice B (818^1) is correct because it properly applies the power rule (multiplying 2 ×\times 3 = 6) then the quotient rule (subtracting 6 - 5 = 1). Choice A (8118^{11}) incorrectly adds 6 + 5 instead of subtracting, choice C (868^6) ignores the division by 8^5, and choice D (8−18^{-1}) would result from 8^5 \div 8^6. Strategy: (1) identify operation (multiply/divide/power of power), (2) check bases match (must be same for properties), (3) apply correct rule (add/subtract/multiply exponents), (4) simplify further (combine like terms, evaluate if needed), (5) handle special cases (zero→\to1, negative\toreciprocal). Common mistakes: multiplying bases when should add exponents, confusing rules (adding when should multiply for powers), working with different bases (23×322^3 \times 3^2 can't simplify using exponent rules).

Question 19

Simplify the expression and write your answer as a single power: (23)4÷25\left(2^3\right)^4 \div 2^5

  1. 2172^{17} (correct answer)
  2. 272^{7}
  3. 2122^{12}
  4. 232^{3}
Explanation: This tests applying integer exponent properties: product rule (am×an=am+na^m \times a^n = a^{m+n}), quotient rule (am÷an=am−na^m \div a^n = a^{m-n}), power rule ((am)n=amn(a^m)^n = a^{m n}), zero exponent (a0=1a^0 = 1), and negative exponents (a−n=1ana^{-n} = \frac{1}{a^n}). For expression ((23)4÷25(2^3)^4 \div 2^5), first apply power rule: ((23)4=23×4=212(2^3)^4 = 2^{3 \times 4} = 2^{12}). Then apply quotient rule: 212÷25=212−5=272^{12} \div 2^5 = 2^{12-5} = 2^7. Choice A (272^7) is correct because it properly applies the power rule (multiplying 3 ×\times 4 = 12) then the quotient rule (subtracting 12 - 5 = 7). Choice B (2172^{17}) incorrectly adds 12 + 5 instead of subtracting, choice C (2122^{12}) ignores the division by 2^5, and choice D (232^3) would result from ((23)1÷20(2^3)^1 \div 2^0). Strategy: (1) identify operation (multiply/divide/power of power), (2) check bases match (must be same for properties), (3) apply correct rule (add/subtract/multiply exponents), (4) simplify further (combine like terms, evaluate if needed), (5) handle special cases (zero →\to 1, negative →\to reciprocal). Common mistakes: multiplying bases when should add exponents, confusing rules (adding when should multiply for powers), working with different bases (23×322^3 \times 3^2 can't simplify using exponent rules).

Question 20

Simplify the expression and write your answer as a single power: (2−3)2÷2−1\left(2^{-3}\right)^2 \div 2^{-1}.

  1. 252^{5}
  2. 2−62^{-6}
  3. 2−72^{-7}
  4. 2−52^{-5} (correct answer)
Explanation: This tests applying integer exponent properties: product rule (am×an=am+na^m \times a^n = a^{m+n}), quotient rule (am÷an=am−na^m \div a^n = a^{m-n}), power rule ((am)n=amna^m)^n = a^{m n}), zero exponent (a0=1a^0 = 1), and negative exponents (a−n=1ana^{-n} = \frac{1}{a^n}). For expression 34×3−23^4 \times 3^{-2}, apply product rule: same base 3, add exponents 4 + (-2) = 2, giving 323^2, then evaluate: 32=93^2 = 9. For (23)42^3)^4, multiply exponents: 3 × 4 = 12 giving 2122^{12}, evaluating: 212=40962^{12} = 4096. Key principle: properties apply when bases are identical; different bases (23×322^3 \times 3^2) can't combine using these rules. For ((2−3)2÷2−1\left(2^{-3}\right)^2 \div 2^{-1}), first apply the power rule: -3 × 2 = -6, giving 2−62^{-6}, then quotient rule: -6 - (-1) = -5, resulting in 2−52^{-5}. Choice B is correct because it properly applies the power rule by multiplying exponents and then the quotient rule by subtracting, handling negatives correctly. Choice A adds when should multiply in power of power ((−3)+2=−1(-3) + 2 = -1 instead of -6). Strategy: (1) identify operation (multiply/divide/power of power), (2) check bases match (must be same for properties), (3) apply correct rule (add/subtract/multiply exponents), (4) simplify further (combine like terms, evaluate if needed), (5) handle special cases (zero → 1, negative → reciprocal). Common mistakes: multiplying bases when should add exponents, confusing rules (adding when should multiply for powers), working with different bases (23×322^3 \times 3^2 can't simplify using exponent rules).