Question

What is the simplest form of the radical expression?
4^sqrt 625x^8y^12

Answer 1

\(\large \bf \sqrt[4]{625x^8y^{12}}\\ \quad \\
\textit{keep in mind that }625=5^4\qquad x^8=x^4\cdot x^4\qquad y^{12}=y^4\cdot y^4\cdot y^4\)

Answer 2

\(\bf \large \sqrt[4]{625x^8y^{12}}\\ \quad \\
\textit{keep in mind that }625=5^4\qquad x^8=x^4\cdot x^4\qquad y^{12}=y^4\cdot y^4\cdot y^4\\ \quad \\
\sqrt[4]{625x^8y^{12}}\implies
\large \sqrt[{\color{red}{ 4}}]{5^4x^4x^4y^4y^4y^4}\)

Answer 3

That doesn't fit any of my choices. My choices are
A. \[5x^2\left| y^3 \right|\]
B. \[25\left| x^2 \right|y^3\]
C. \[5x^4\left| y^3 \right|\]
D. \[25\left| x^4y^3 \right|\]

Answer 4

A

Answer 5

Yes, because we don't know the values of x and y, we can't do the simplification of y^12 quite so straightforwardly because y^3 will have a different value if y is negative.

Answer 6

Thank you!

Answer 7

For example, if we use y=1, x=1 for ease of arithmetic, the original expression works out to be 5. Same for y=-1, x=1. If we simplify it to 5x^2y^3 as one might do when unaware of this wrinkle, with y=1,x=1 we still get 5, but with y=-1,x=1, we get -5! Our simplification would be incorrect if \(y<0\). That's why writing it as \(5x^2|y^3|\) is done.