Is My Solution to the Complex Algebraic Expression Correct?

[mathdrama]

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.

 

[Opalg]

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.

 

[soroban]

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: .$$\left(\tfrac{a}{b}\right)^{-1} \:=\:\tfrac{b}{a}$$

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

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

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

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

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

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

. . $$=\;\frac{4x^2}{y}$$

 

[sweer6]

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.

 

[CellCoree]

First off, I want to commend you for taking the initiative to evaluate this equation on your own. It shows that you are actively trying to understand and learn, which is a great quality to have as a scientist.

Now, let's take a closer look at your solution and see if it is correct. The first step you took was to simplify the numerator and denominator separately. This is a good strategy and it looks like you correctly simplified the numerator to 32^2/5x^2y^8/5 and the denominator to (4x^2)^(2y^3/5).

Next, you took the reciprocal of the denominator and brought it to the numerator. This is also correct, but instead of using a negative exponent, you could have simply rewritten the denominator as (4x^2)^(-2y^3/5).

From there, you correctly simplified the expression further by taking the fifth root of 32 and squaring it to get 4. However, when you brought the denominator to the numerator, you should have multiplied the exponents instead of adding them. So the final simplified expression should be 4x^2y^(-8/5) / (4x^2)^(-2y^3/5).

I hope this helps clarify any confusion and good job on attempting to solve this equation! Keep up the good work.