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

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

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Answer 1 · 2018-05-24T08:30:23

$\frac{3}{4}$ $3/4$

Explanation:

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

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

$\Rightarrow x^{0} = 1$ $color(brown)(rArrx^0=1$

$\Rightarrow x^{- z} = \frac{1}{x^{z}}$ $color(brown)(rArrx^-z=1/(x^z)$

$\Rightarrow x^{\frac{1}{y}} = \sqrt[y]{x}$ $color(brown)(rArrx^(1/y)=root(y)(x)$

Now, insert $8$ $8$ into the problem

$\to {((8^{- 2}) (27 \cdot 8^{0}))}^{\frac{1}{3}}$ $rarr((8^-2)(27*8^0))^(1/3)$

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

$\to \to {((8^{- 2}) (27))}^{\frac{1}{3}}$ $rarrrarr((8^-2)(27))^(1/3)$

Apply $x^{- z} = \frac{1}{x^{z}}$ $color(brown)(x^-z=1/x^z$

$\to {((\frac{1}{8^{2}}) (27))}^{\frac{1}{3}}$ $rarr((1/8^2)(27))^(1/3)$

$\to {((\frac{1}{64}) (27))}^{\frac{1}{3}}$ $rarr((1/64)(27))^(1/3)$

$\to {(\frac{27}{64})}^{\frac{1}{3}}$ $rarr(27/64)^(1/3)$

Apply $x^{\frac{1}{y}} = \sqrt[y]{x}$ $color(brown)(x^(1/y)=root(y)(x)$

$\to \sqrt[3]{\frac{27}{64}}$ $rarrroot(3)(27/64)$

$\to \sqrt[3]{\frac{3 \times 3 \times 3}{4 \times 4 \times 4}}$ $rarrroot(3)((3xx3xx3)/(4xx4xx4))$

$\Rightarrow \frac{3}{4}$ $color(green)(rArr3/4$

Hope that helps!!! ☻

Answer 2 · 2018-05-24T08:31:25

$\frac{3}{4}$ $3/4$

Explanation:

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

When $x = 8$ $x=8$ , we get:

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

$= {(\frac{1}{8^{2}} \cdot 27 \cdot 1)}^{\frac{1}{3}}$ $=(1/8^2*27*1)^(1/3)$

$= {(\frac{1}{64} \cdot 27)}^{\frac{1}{3}}$ $=(1/64*27)^(1/3)$

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

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

$= \frac{\sqrt[3]{27}}{\sqrt[3]{64}}$ $=(root(3)27)/(root(3)64)$

$= \frac{3}{4}$ $=3/4$

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

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