Ok so I need to simplify this:
$$\sqrt[\LARGE 6] {x^{25}y^{10}\over x^7 y^4}$$
I can easily break it down to the answer $x^3 y$ but I have seen that my answer is actually meant to be $|x^3 y|$
Can anyone please explain to me why I must put my answer in an absolute value?
My question instructs me to simplify the expression and assume all variables are positive.
Thank you
Because $x^3 y$ can be negative, but the expression cannot. (E.g. try $x=-1$ and $y=1$ and see that the results don't match.)
In the expression $\frac{x^{25} y^{10}}{x^7 y^4}$, (a) both $y^4$ and $y^{10}$ are non-negative regardless of the value of $y \in \mathbb{R}$, and (b) $x^{25}/x^7=x^{18}$ is non-negative regardless of the value of $x \in \mathbb{R}$. So we're taking the $6$-th root of a non-negative real number.
Similar to the situation with $x^2=(-x)^2$ for real $x$, we have $x^6=(-x)^6$ for real $x$ (along with complex roots). We define $\sqrt[6]{\text{[this]}}$ as returning the positive real $6$-th root.
Think of the simple example $y=\sqrt{x^2}$ and plot it: You'll get $|x|$, since negative values for $x$ get positive in anyway...
