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Simplify the expression: ((243)⁻²)/(5) x (125)^(2)/(3)(9)-(3)/(2)

Mathematics

ScanSolve AI Answer

Answer

75

Step-by-step solution

7 steps

  1. Express the bases as powers of prime numbers.

    We have 243=35243 = 3^5, 125=53125 = 5^3, and 9=329 = 3^2.

  2. Substitute these into the expression.

    (35)−25×(53)23(32)−32\frac{(3^5)^{-\frac{2}{5}} \times (5^3)^{\frac{2}{3}}}{(3^2)^{-\frac{3}{2}}}

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

    For the numerator:
    (35)−25=35×(−25)=3−2(3^5)^{-\frac{2}{5}} = 3^{5 \times (-\frac{2}{5})} = 3^{-2}
    (53)23=53×(23)=52(5^3)^{\frac{2}{3}} = 5^{3 \times (\frac{2}{3})} = 5^2

    For the denominator:
    (32)−32=32×(−32)=3−3(3^2)^{-\frac{3}{2}} = 3^{2 \times (-\frac{3}{2})} = 3^{-3}

  4. Rewrite the expression with the simplified terms.

    3−2×523−3\frac{3^{-2} \times 5^2}{3^{-3}}

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

    3−2−(−3)×523^{-2 - (-3)} \times 5^2 3−2+3×523^{-2 + 3} \times 5^2 31×523^1 \times 5^2

  6. Calculate the values.

    31=33^1 = 3
    52=255^2 = 25

  7. Multiply the results.

    3×25=753 \times 25 = 75

    The simplified expression is:

    7575
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