TITLE: How do I reduce the index of a radical $\sqrt[9]{64a^6b^{12}}$?
QUESTION [0 upvotes]: $\sqrt[9]{64a^6b^{12}}$
It doesn't have to be the answer, just tell me what I should do

REPLY [1 votes]: Notice that, $\sqrt[n]{x}=x^{\frac{1}{n}}$, in your case, $n = 9$, and $64=2^6$, so
$$\sqrt[9]{64a^6b^{12}}=\sqrt[9]{2^6a^6b^{12}}=\sqrt[9]{(2a)^6b^{12}}$$
Using the above property,
$$\sqrt[9]{(2a)^6b^{12}}=\Big((2a)^6b^{12}\Big)^{\frac{1}{9}}=(2a)^{\frac{6}{9}}b^{\frac{12}{9}}=(2a)^{\frac{2}{3}}b^{\frac{4}{3}}=\Big((2a)^2b^{4}\Big)^{\frac{1}{3}}$$
Therefore,
$$\sqrt[9]{(2a)^6b^{12}}=\Big((2a)^2b^{4}\Big)^{\frac{1}{3}}=\sqrt[3]{(2a)^2b^{4}}$$
It is possible to simplify a little more, noting that $b^4=b\cdot b^3$, so
$$\sqrt[3]{(2a)^2b^{4}}=\sqrt[3]{(2a)^2b\cdot b^3}=b\sqrt[3]{(2a)^2b}$$
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