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Math Help - Simplifying Radicals and Exponents Problem

  1. #1
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    Question Simplifying Radicals and Exponents Problem

    How would you solve:

    ((25x^3)^(2/3))) / ((125xy^2)^(1/3)))

    ?

    I got the cube root of 625x^6 in the numerator and the cube root of 125xy^2 in the denominator.
    After I simplified them, I got 5x^2 * (cube root of 5) in the numerator and 5(xy^2)^(1/3) in the denominator, but that doesn't look right to me.

    Please help!
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  2. #2
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    \frac{(25x^3)^{2/3}}{(125xy^2)^{1/3}} = \frac{\sqrt[3]{125 \cdot 5}x^2}{5\sqrt[3]{x}y^{2/3}} = \boxed{\sqrt[3]{5}x^{5/3}y^{-2/3}}

    That seems good to me but I'm kind of tired.
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  3. #3
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    Hello, MathGeek06!

    TrevorP is absolutely correct . . .


    Simplify: . \frac{(25x^3)^{\frac{2}{3}}} {(125xy^2)^{\frac{1}{3}}}
    I would do it like this . . .

    . . \frac{\left(5^2x^3\right)^{\frac{2}{3}}}{\left(5^3  xy^2\right)^{\frac{1}{3}}} \;=\;\frac{(5^2)^{\frac{2}{3}}(x^3)^{\frac{2}{3}}} {(5^3)^{\frac{1}{3}}(x^{\frac{1}{3}})(y^2)^{\frac{  1}{3}}} \;=\; \frac{5^{\frac{4}{3}}x^2} {5^1x^{\frac{1}{3}}y^{\frac{2}{3}} } \;=\;\boxed{\frac{5^{\frac{1}{3}}x^{\frac{5}{3}}}{  y^{\frac{2}{3}}}}

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