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Simplify the expression: ((33)2/3 ext div 24) / (25)⁻³/5

Mathematics

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Answer

92\frac{9}{2}

Step-by-step solution

8 steps

  1. Express the bases as powers of prime numbers.

    We have 27=3327 = 3^3 and 32=2532 = 2^5.

  2. Substitute these into the expression.

    (33)23÷24(25)−35\frac{(3^3)^{\frac{2}{3}} \div 2^4}{(2^5)^{-\frac{3}{5}}}

  3. Apply the power rule (am)n=amn(a^m)^n = a^{mn} to simplify the terms.

    For the numerator:
    (33)23=33×23=32(3^3)^{\frac{2}{3}} = 3^{3 \times \frac{2}{3}} = 3^2

    For the denominator:
    (25)−35=25×(−35)=2−3(2^5)^{-\frac{3}{5}} = 2^{5 \times (-\frac{3}{5})} = 2^{-3}

  4. Rewrite the expression with the simplified terms.

    32÷242−3\frac{3^2 \div 2^4}{2^{-3}}

  5. Calculate the values of the powers.

    32=93^2 = 9
    24=162^4 = 16
    2−3=123=182^{-3} = \frac{1}{2^3} = \frac{1}{8}

  6. Substitute these values back into the expression.

    9÷1618\frac{9 \div 16}{\frac{1}{8}} 91618\frac{\frac{9}{16}}{\frac{1}{8}}

  7. Perform the division of fractions.

    To divide by a fraction, multiply by its reciprocal.

    916×81\frac{9}{16} \times \frac{8}{1} 9×816×1\frac{9 \times 8}{16 \times 1} 7216\frac{72}{16}

  8. Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 8.

    72÷816÷8=92\frac{72 \div 8}{16 \div 8} = \frac{9}{2}

    The simplified expression is:

    92\frac{9}{2}
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