I'm trying to write $(\sqrt{3}/\sqrt{6})^3$ to simplest form possible which should be $1/4\sqrt{2}$.
This is what I tried: (√3/√6)^3 = (√3/√6 * √6/√6)^3 = (√18/6)^3 = √18^3/6^3 = √5832 / 216 =... I'm kinda stuck here and do not know what mathematics rule I have to apply.
Could you help me with showing where my miscalculation is and helping me with step by step?
Edit: I'm also new to stackexchange so I have no idea if i provided enough information
The three things to note are:
$(\sqrt{a})^3 = (\sqrt{a})^2 \sqrt {a} = a*\sqrt{a}$.
And:
$\sqrt{a*b} = \sqrt{a}*\sqrt{b}$ (assuming $a$ and $b$ are both positive.)
And, obviously:
$(\frac ab)^k = \frac {a^k}{b^k}$.
So:
$(\frac {\sqrt 3}{\sqrt 6})^3 = (\frac {\sqrt 3}{\sqrt 2*\sqrt 3})^3 = (\frac 1{\sqrt 2})^3 = \frac 1{(\sqrt 2)^3} = \frac 1{2\sqrt 2}$.
And if you have one of those textbooks that insist you can't have a radical in then denominator:
$\frac 1{2\sqrt 2}= \frac {\sqrt{2}}{2\sqrt{2}\sqrt{2}} = \frac {\sqrt{2}}4$.
...
To finish what you began:
$ (√3/√6)^3 = (√3/√6 * √6/√6)^3 = (√18/6)^3 = √18^3/6^3 = √5832 / 216 =$
$\frac{\sqrt{5832}}{216} = \frac{\sqrt {54^2*2}}{216}= \frac {54\sqrt{2}}{216}=\frac {\sqrt 2}{4}$.
Which should be a lesson to you to simplify as you go along. It'd have been easier to do:
$(\frac {\sqrt{18}}{6})^2 =( \frac {\sqrt{9*2}}{6})^3 = (\frac {3*\sqrt 2}{6})^3 = (\frac {\sqrt 2}2)^3 = \frac {\sqrt{2^3}}{2^3} = \frac {\sqrt{2^2*2}}{2^3} = \frac {2 \sqrt 2}{2^3} = \frac {\sqrt 2}{2^2} =\frac {\sqrt 2}{4}$.
If you've learned this there is alson $\sqrt[k]{a} = a^{\frac 1k}$.
So $(\frac {\sqrt 3}{\sqrt 6})^3 = $$(\frac {3^{\frac 12}}{6^{\frac 12}})^3 $$= \frac {3^{\frac 32}}{6^{\frac 32}} = (\frac 36)^{1\frac 12}=(\frac 12)^{1\frac 12} =\frac 12*\sqrt {\frac 12} = \frac 1{2\sqrt{2}}$.
Moral: In Math, there are more than one way to do things.