Question

How do you simplify `x^-4/(2xy^3*(2x^3y^2)^3)`?

Answer 1

`1/(16x^14y^9)`

Explanation:

Before we start note that `2` is the same as `2^1`

`color(blue)("Consider "x^(-4))`

`x^(-4)` is the same as `1/x^4`

As an additional point: `1/x^(-4)` is the same as `x^4`
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`color(blue)("Consider "(2x^3y^2)^3`

This is the same as `(2x^3y^2)xx(2x^3y^2)xx(2x^3y^2)`

Giving:`" "8x^9y^6`

So as a general case:`" "(x^ay^b)^c = x^(axxc)y^(bxxc)`
So for `(2x^3y^2)^3 = 2^(1xx3)x^(3xx3)y^(2xx3)`

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`color(blue)("Putting it all together")`
`x^-4/(2xy^3(2x^3y^2)^3) = 1/x^4xx 1/(2xy^3(2x^3y^2)^3)=1/x^4xx1/(2xy^3xx8x^9y^6)`

`" "=1/(16x^14y^9)`