Question

How do you simplify `(20x^5)/y^2 * ((x^2y^2)/(2x))^3`?

Answer 1

`5/2 x^8 y^4`

Explanation:

First of all, expand the power of the second factor: since the power of a fraction is the power of the numerator divided by the power of the denominator, we have that

`({x^2y^2}/{2x})^3=(x^2y^2)^3/(2x)^3`

Now, the power of a product is again the power of every single factor, so

`(x^2y^2)^3= (x^2)^3 * (y^2)^3`, and `(2x)^3 = 2^3 * x^3`.

The last rule we need is the one which states that when we deal with the power of a power, we must multiplicate the exponents:

`(x^2)^3 = x^{2*3)=x^6`, and the same goes for `(y^2)^3`

The result is thus

`({x^2y^2}/{2x})^3 = (x^6y^6)/(8x^3)`.

Now we're ready to multiply and cross simplify:

`(color(green)(20)x^5)/color(red)(y^2) * (color(blue)(x^6)color(red)(y^6))/(color(green)(8)color(blue)(x^3))`

So, we can simplify `20` and `8` dividing both by `4`, obtaining `5` and `2`.

Also, `y^2` cancels out and `y^6` becomes `y^4`, and `x^3` cancels out too and `x^6` becomes `x^3`.

So, we're left with

`5x^5 * (x^3y^4)/2= (5x^8 y^4)/2`