I've been self-studying from Stroud & Booth's amazing "Engineering Mathematics", and am on the (precalculus) Algebra section. I'm trying to simplify this polynomial, and I've tried it a 101 ways, but I just can't seem to get a reasonable result. The polynomial is:
$$\sqrt[3]{x^9y^\frac{1}{3}z^\frac{1}{2}}\times{y}^\frac{8}{9}\times(2^{-8}\times{x}^6\times{y}^2\times{z}^\frac{1}{3})^\frac{-1}{2}$$
Any help would be greatly appreciated.
Noting that $\sqrt[m]{a^n}=a^{\frac{n}{m}} $ and $a^{-n}=\frac{1}{a^n}$, we have
$$\sqrt[3]{x^9y^\frac{1}{3}z^\frac{1}{2}}\times{y}^\frac{8}{9}\times(2^{-8}\times{x}^6\times{y}^2\times{z}^\frac{1}{3})^\frac{-1}{2}$$
$$=x^{3}y^{\frac{1}{9}} z^{\frac{1}{6}}\times y^{\frac{8}{9}}\times\frac{1}{\sqrt{2^{-8}\times{x}^6\times{y}^2\times{z}^\frac{1}{3}}}$$
$$=x^3y z^{\frac{1}{6}}\times\frac{2^4}{x^3yz^{\frac{1}{6}}}=2^4=16 $$