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How would I simplify the following radical.

$$\large \sqrt[12]{64a^{3}b^{6}}$$

I know that I can do

$$\large 64^{\frac{1}{12}}a^{\frac{3}{12}}b^{\frac{6}{12}}$$

But I am unsure how to simplify it further.

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up vote 3 down vote accepted

$$\sqrt[\large 12]{64a^{3}b^{6}}= (64)^{1/12}a^{3/12}b^{6/12} = (2^6)^{1/12}a^{1/4}b^{1/2} = 2^{6/12}a^{1/4}b^{1/2} = 2^{1/2}a^{1/4}b^{1/2} = (2b)^{1/2}a^{1/4}$$

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I see so my final answer would be square root(2b) 4 root(a) – Fernando Martinez Jan 17 '13 at 17:23
Yes, exactly! $\sqrt{2b}\cdot \sqrt[\large 4]{a}$ which can also be expressed as $\sqrt[\large 4]{4b^2a}$ – amWhy Jan 17 '13 at 17:27

Once you are able to translate radicals in powers with fractional exponents, then all you have to do is to exploit fractions' and powers' properties. This way you can prove all the fundamental properties regarding products and quotients of radicals.

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