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Use the exponent rule a^m x aⁿ = a^m+n.

Mathematics

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2 parts
a

14\frac{1}{4}

b

9128\frac{9}{128}

Step-by-step solution

9 steps

Here's how to simplify the expressions:

Part a· 2 steps

Simplify 42×4−34^2 \times 4^{-3}

  1. Use the exponent rule am×an=am+na^m \times a^n = a^{m+n}.

    42×4−3=42+(−3)4^2 \times 4^{-3} = 4^{2 + (-3)} 42+(−3)=42−34^{2 + (-3)} = 4^{2-3} 42−3=4−14^{2-3} = 4^{-1}

  2. Convert the negative exponent to a positive exponent using the rule a−n=1ana^{-n} = \frac{1}{a^n}.

    4−1=1414^{-1} = \frac{1}{4^1} 4−1=144^{-1} = \frac{1}{4}

    The simplified expression is 14\boxed{\frac{1}{4}}.

Part b

9128\frac{9}{128}

Simplify 32(22)−223\frac{3^2 (2^2)^{-2}}{2^3}

  1. Simplify the term (22)−2(2^2)^{-2} in the numerator using the exponent rule (am)n=amn(a^m)^n = a^{mn}.

    (22)−2=22×(−2)(2^2)^{-2} = 2^{2 \times (-2)} (22)−2=2−4(2^2)^{-2} = 2^{-4}

  2. Substitute this back into the expression.

    32×2−423\frac{3^2 \times 2^{-4}}{2^3}

  3. Simplify the terms with base 2 using the exponent rule aman=am−n\frac{a^m}{a^n} = a^{m-n}.

    2−423=2−4−3\frac{2^{-4}}{2^3} = 2^{-4-3} 2−4−3=2−72^{-4-3} = 2^{-7}

  4. Combine the simplified terms.

    32×2−73^2 \times 2^{-7}

  5. Convert the negative exponent to a positive exponent using the rule a−n=1ana^{-n} = \frac{1}{a^n}.

    2−7=1272^{-7} = \frac{1}{2^7}

  6. Write the expression with positive exponents.

    32×127=32273^2 \times \frac{1}{2^7} = \frac{3^2}{2^7}

  7. Calculate the values of the powers.

    32=93^2 = 9 27=1282^7 = 128

    The simplified expression is 9128\boxed{\frac{9}{128}}.

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