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Apply the negative exponent rule ((a)/(b))^-n = ((b)/(a))ⁿ.

Mathematics

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Answer

781252\frac{78125}{2}

Step-by-step solution

7 steps

  1. Apply the negative exponent rule (ab)−n=(ba)n\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n.

    (45)2×54×(52)2÷(25)3\left(\frac{4}{5}\right)^2 \times 5^4 \times \left(\frac{5}{2}\right)^2 \div \left(\frac{2}{5}\right)^3

  2. Apply the exponent rule (ab)n=anbn\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} and express 4 as 222^2.

    4252×54×5222÷2353\frac{4^2}{5^2} \times 5^4 \times \frac{5^2}{2^2} \div \frac{2^3}{5^3} (22)252×54×5222÷2353\frac{(2^2)^2}{5^2} \times 5^4 \times \frac{5^2}{2^2} \div \frac{2^3}{5^3} 2452×54×5222÷2353\frac{2^4}{5^2} \times 5^4 \times \frac{5^2}{2^2} \div \frac{2^3}{5^3}

  3. Convert the division to multiplication by taking the reciprocal of the last term.

    2452×54×5222×5323\frac{2^4}{5^2} \times 5^4 \times \frac{5^2}{2^2} \times \frac{5^3}{2^3}

  4. Group terms with the same base and apply the exponent rules am×an=am+na^m \times a^n = a^{m+n} and aman=am−n\frac{a^m}{a^n} = a^{m-n}.

    For base 2:

    2422×23=2422+3=2425=24−5=2−1\frac{2^4}{2^2 \times 2^3} = \frac{2^4}{2^{2+3}} = \frac{2^4}{2^5} = 2^{4-5} = 2^{-1}

    For base 5:

    54×52×5352=54+2+352=5952=59−2=57\frac{5^4 \times 5^2 \times 5^3}{5^2} = \frac{5^{4+2+3}}{5^2} = \frac{5^9}{5^2} = 5^{9-2} = 5^7

  5. Combine the simplified terms.

    2−1×57=12×572^{-1} \times 5^7 = \frac{1}{2} \times 5^7

  6. Calculate 575^7.

    57=5×5×5×5×5×5×5=781255^7 = 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 = 78125

  7. Substitute the value and simplify.

    12×78125=781252\frac{1}{2} \times 78125 = \frac{78125}{2}

    The final answer is 781252\frac{78125}{2}.

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