When x=8, what is the value of ((x^-2)(27x^0))^(1/3)?

May 24, 2018

$\frac{3}{4}$

Explanation:

color(blue)(((x^-2)(27x^0))^(1/3) $\text{when}$ color(brown)(x=8

To, solve this, we need to know some exponential rules

color(brown)(rArrx^0=1

color(brown)(rArrx^-z=1/(x^z)

color(brown)(rArrx^(1/y)=root(y)(x)

Now, insert $8$ into the problem

$\rightarrow {\left(\left({8}^{-} 2\right) \left(27 \cdot {8}^{0}\right)\right)}^{\frac{1}{3}}$

Now, apply color(brown)(x^0=1

$\rightarrow \rightarrow {\left(\left({8}^{-} 2\right) \left(27\right)\right)}^{\frac{1}{3}}$

Apply color(brown)(x^-z=1/x^z

$\rightarrow {\left(\left(\frac{1}{8} ^ 2\right) \left(27\right)\right)}^{\frac{1}{3}}$

$\rightarrow {\left(\left(\frac{1}{64}\right) \left(27\right)\right)}^{\frac{1}{3}}$

$\rightarrow {\left(\frac{27}{64}\right)}^{\frac{1}{3}}$

Apply color(brown)(x^(1/y)=root(y)(x)

$\rightarrow \sqrt[3]{\frac{27}{64}}$

$\rightarrow \sqrt[3]{\frac{3 \times 3 \times 3}{4 \times 4 \times 4}}$

color(green)(rArr3/4

Hope that helps!!! ☻

May 24, 2018

$\frac{3}{4}$

Explanation:

Given: ${\left({x}^{-} 2 \cdot 27 {x}^{0}\right)}^{\frac{1}{3}}$

When $x = 8$, we get:

$= {\left({8}^{-} 2 \cdot 27 \cdot {8}^{0}\right)}^{\frac{1}{3}}$

$= {\left(\frac{1}{8} ^ 2 \cdot 27 \cdot 1\right)}^{\frac{1}{3}}$

$= {\left(\frac{1}{64} \cdot 27\right)}^{\frac{1}{3}}$

$= {\left(\frac{27}{64}\right)}^{\frac{1}{3}}$

$= \frac{{27}^{\frac{1}{3}}}{{64}^{\frac{1}{3}}}$

$= \frac{\sqrt[3]{27}}{\sqrt[3]{64}}$

$= \frac{3}{4}$