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  1. #1
    Senior Member euclid2's Avatar
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    simplify

    This will most likely be hard to interpret, I apologize.

     (3 \sqrt{(27x^6y^14)}^2/ \sqrt{(9x^2y^6)}^3)^-1

    Note: the fraction is in brackets, therefore it is all to the power of -1.
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  2. #2
    Member Greengoblin's Avatar
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    Do you mean \sqrt{(27x^6y^{14})^2} and \sqrt{(9x^2y^6)^3} , or \left[\sqrt{(27x^6y^{14})}\right]^2 and \left[\sqrt{(9x^2y^6)}\right]^3?
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  3. #3
    Senior Member euclid2's Avatar
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    Quote Originally Posted by Greengoblin View Post
    Do you mean \sqrt{(27x^6y^{14})^2} and \sqrt{(9x^2y^6)^3} , or \left[\sqrt{(27x^6y^{14})}\right]^2 and \left[\sqrt{(9x^2y^6)}\right]^3?
    \left[3\sqrt{(27x^6y^{14})}\right]^2 and \left[\sqrt{(9x^2y^6)}\right]^3

    Note the 3 before the sqrt in the numerator and all is to the power of -1, such that the ^2 in the numerator and the ^3 in the denominator are inside the brackets.
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  4. #4
    Member Greengoblin's Avatar
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    Ok well first the minus 1 exponent in the outer brackets tells you to take the inverse of the fraction. i.e:

    \left(\frac{\left[3 \sqrt{27x^6y^{14}}\right]^2}{\left[\sqrt{9x^2y^6}\right]^3}\right)^{-1}=\frac{\left[\sqrt{9x^2y^6}\right]^3}{\left[3\sqrt{27x^6y^{14}}\right]^2}

    next, you can cancel the square root on the denominator, and the numerator exponent becomes 3/2, since the square root can be replaced by the exponent, 1/2, and we know as a rule of indicies,
    (a^m)^n=a^{mn}....

    \frac{\left[9x^2y^6\right]^{3/2}}{3^227x^6y^{14}}

    Next step is to use this rule again on the numerator:

    \frac{9^{3/2}x^{6/2}y^{18/2}}{243x^6y^{14}}=\frac{27x^3y^9}{243x^6y^{14}}=\f  rac{1}{9x^3y^5}

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  5. #5
    Senior Member euclid2's Avatar
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    Quote Originally Posted by Greengoblin View Post
    Ok well first the minus 1 exponent in the outer brackets tells you to take the inverse of the fraction. i.e:

    \left(\frac{\left[3 \sqrt{27x^6y^{14}}\right]^2}{\left[\sqrt{9x^2y^6}\right]^3}\right)^{-1}=\frac{\left[\sqrt{9x^2y^6}\right]^3}{\left[3\sqrt{27x^6y^{14}}\right]^2}

    next, you can cancel the square root on the denominator, and the numerator exponent becomes 3/2, since the square root can be replaced by the exponent, 1/2, and we know as a rule of indicies,
    (a^m)^n=a^{mn}....

    \frac{\left[9x^2y^6\right]^{3/2}}{3^227x^6y^{14}}

    Next step is to use this rule again on the numerator:

    \frac{9^{3/2}x^{6/2}y^{18/2}}{243x^6y^{14}}=\frac{27x^3y^9}{243x^6y^{14}}=\f  rac{1}{9x^3y^5}

    I really appreciate the explanations, it helps very much.
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