Question

Simplify $\sqrt[4]{81x^8{y}^4{z}^{16}}$.

Answer 1

$$\begin{gathered} \sqrt[4]{81x^8{y}^4{z}^{16}} \\ ={\left(3^4\times x^8\times y^4\times z^{16}\right)}^{\frac{1}{4}} \\ ={\left(3^4\right)}^{\frac{1}{4}}\times {\left(x^8\right)}^{\frac{1}{4}}\times {\left(y^4\right)}^{\frac{1}{4}}\times {\left(z^{16}\right)}^{\frac{1}{4}}\left[{\left(x\times y\times z\times ...\right)}^a=x^a\times y^a\times z^a\times ...\right] \end{gathered}$$
$$\begin{gathered} =\left(3^{4\times \frac{1}{4}}\right)\times \left(x^{8\times \frac{1}{4}}\right)\times \left(y^{4\times \frac{1}{4}}\right)\times \left(z^{16\times \frac{1}{4}}\right)\left[{\left(x^a\right)}^b=x^{ab}\right] \\ =3\times x^2\times y\times z^4 \\ =3x^{2}y{z}^4 \end{gathered}$$