Question

How do you simplify `sqrt(x^3y^9)`?

Answer 1

First of all, use the fact that `\sqrt{a\cdot b}=\sqrt{a}\sqrt{b}`. So, we have that `\sqrt{x^3y^9}=\sqrt{x^3}\sqrt{y^9}`

Let's deal with a root at a time: we can write `x^3` as `x^2\cdot x`. So, `\sqrt{x^3}=\sqrt{x^2\cdot x}=\sqrt{x^2}\sqrt{x}=x\sqrt{x}`.

For the same reasons, we write `y^9` as `y^2\cdoty^2\cdoty^2\cdoty^2\cdot y`. So, we have that `\sqrt{y^9}=\sqrt{y^2\cdoty^2\cdoty^2\cdoty^2\cdot y}=\sqrt{y^2}\sqrt{y^2}\sqrt{y^2}\sqrt{y^2}\sqrt{y}`, which equals `y^4\sqrt{y}`.

Putting the two things together, we have that
`\sqrt{x^3y^9} = xy^4\sqrt{xy}`.