MHB Is My Solution to the Complex Algebraic Expression Correct?

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The discussion focuses on evaluating the complex algebraic expression [32x^5y^4)^2/5/(4x^2)^2y^3/5]^-1. Participants clarify that the expression should be simplified by first inverting the fraction due to the negative exponent. The correct approach involves rewriting the expression and applying exponent rules to simplify the components step by step. Ultimately, the simplified result is determined to be 4x^2/y. Confusion arises regarding the placement of the exponent and its application to the entire expression versus individual components.
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I had to evaluate this: [32x^5y^4)^2/5/(4x^2)^2y^3/5]^-1

This is how I did it, but is it correct?

= [32^2/5x^2y^4(2/5)-1
(4x^2)^(2y^3/5)

= [32^2/5x^2y^8/5)^-1
(4x2)^(2y^3/5)

= [(^5√32)^2*x^2y^8/5)-1
(4x2)^(2y^3/5)

= [(2)^2*x^2y^8/5)-1
(4x^2)^(2y^3/5)

= [4*x^2y^8/5]^-1
(4x^2)^(2y^3/5)= [4x^2y^8/5]-1
(4x^2)^(2y^3/5)= (4x^2)(^2y^3/5)
4x^2y^8/5
I apologize for all the messiness.
 
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Apart from all the messiness (of which there is a lot), I think that you may be getting towards the right answer. The final line of the calculation, $\dfrac{\text{(4x^2)(^2y^3/5)}}{\text{4x^2y^8/5}}$, makes no sense at all as it stands. But if you put one of the parentheses in the top row in a different position, so that it reads $\dfrac{\text{(4x^2)^2 y^(3/5)}} {\text{4x^2y^8/5}} = \dfrac{(4x^2)^2 y^{3/5}}{4x^2y^{8/5}}$, then you are on the right lines. But you can still simplify this further, by cancelling some of the stuff in the numerator with parts of the denominator.
 
Hello, mathdrama!

I think your problem looks like this:
\text{Simplify: }\:\left[\frac{(32x^5y^4)^{\frac{2}{5}}}{(4x^2)^2y^{\frac{3}{5}}}\right]^{-1}
A fraction to the minus-one power is inverted: .\left(\tfrac{a}{b}\right)^{-1} \:=\:\tfrac{b}{a}

\left[\frac{(32x^5y^4)^{\frac{2}{5}}}{(4x^2)^2y^{\frac{3}{5}}}\right]^{-1}

. . =\;\frac{(2^2\cdot x^2)^2\cdot y^{\frac{3}{5}}}{(2^5\cdot x^5\cdot y^4)^{\frac{2}{5}}}

. . =\;\frac{(2^2)^2\cdot (x^2)^2\cdot y^{\frac{3}{5}}} {(2^5)^{\frac{2}{5}}\cdot (x^5)^{\frac{2}{5}}\cdot (y^4)^{\frac{2}{5}}}

. . =\;\frac{2^4\cdot x^4\cdot y^{\frac{3}{5}}}{2^2\cdot x^2\cdot y^{\frac{8}{5}}}

. . =\;2^{4-2}\cdot x^{4-2}\cdot y^{\frac{3}{5}-\frac{8}{5}}

. . =\;2^2\cdot x^2\cdot y^{-1}

. . =\;\frac{4x^2}{y}
 
what i quite don't understand about this question "[32x^5y^4)^2/5/(4x^2)^2y^3/5]^-1" is after the first raise to the power of two (bold above) is the index devided by 5 and so on or is it the question so far that is divided by 5 and so on.
 
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