# Simplifying roots

Lets say I have this: $2\dfrac{2}{3}\sqrt{2\dfrac{2}{3}}$ and I want to make this more simple, are my following steps correct?

$2\dfrac{2}{3}\sqrt{2\dfrac{2}{3}} = 2\dfrac{2}{3}\sqrt{\dfrac{24}{9}}$ = $2\dfrac{2}{3} . \dfrac{1}{3} . 2\sqrt{6} = \dfrac{16}{9} \sqrt{6}$

-
Looks good to me, provided that $2{2\over3}$ means two-and-two-thirds, and not two-times-two thirds. – Gerry Myerson Oct 13 '12 at 11:23
Thank you, it indeed means two-and-two-thirds – Tittyboy Oct 13 '12 at 11:26

$$2\frac{2}{3}\sqrt{2\frac{2}{3}}=\frac{8}{3}\sqrt{\frac{8}{3}}=\frac{16}{3}\sqrt{\frac{2}{3}}=\frac{16}{9}\sqrt{6}$$
The second step is wrong, since $2 \cdot 2 \cdot 3 = 12$, not $24$. I think you should note that $\sqrt{2 \cdot 2} = 2$.
I mean, what is in the root is actually $\frac{4}{3}$, and you know that the root of a quotient is the quotient of the roots. Then, the only thing you gotta do is rationalise the result. – busman Oct 13 '12 at 11:18
What busman means is that $2\cdot \frac{2}{3}$ is $\frac{12}{9}$. But as pointed out above, $2\frac{2}{3}$ in this case means $2+\frac{2}{3} = \frac{8}{3} = \frac{24}{9}$ – Arthur Oct 13 '12 at 11:29