Question

Simplify 8 power 1/3 * 16power 1/3 / 32 power -1/3

Answer 1

Answer:

$Value \: of \:\frac{8^{\frac{1}{3}}\times (16)^{\frac{1}{3}}}{(32)^{\frac{-1}{3}}}=16$

Step-by-step explanation:

$\frac{8^{\frac{1}{3}}\times (16)^{\frac{1}{3}}}{(32)^{\frac{-1}{3}}}$

$=8^{\frac{1}{3}}\times (16)^{\frac{1}{3}}\times (32)^{\frac{-1}{3}}$

/* By Exponential identity:

$\boxed { \frac{1}{a^{-n}}=a^{n}}$

$=\sqrt[3]{8\times 16 \times 32}$

$=\sqrt[3]{2^{3}\times 2^{4}\times 2^{5}}$

$=\sqrt[3]{2^{3+4+5}}$

$=\sqrt[3]{2^{12}}$

$= 2^{\frac{12}{3}}$

/* $\boxed {(a^{m})^{n}=a^{mn}}$*/

$=2^{4}$

$=16$

Therefore,

$Value \: of \:\frac{8^{\frac{1}{3}}\times (16)^{\frac{1}{3}}}{(32)^{\frac{-1}{3}}}=16$

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Answer 2

Given:

The expression is

$\dfrac{8^{\frac{1}{3}}\times 16^{\frac{1}{3}}}{32^{-\frac{1}{3}}}$

To find:

The simplified form of the given expression.

Solution:

We have,

$\dfrac{8^{\frac{1}{3}}\times 16^{\frac{1}{3}}}{32^{-\frac{1}{3}}}$

$=\dfrac{(2^3)^{\frac{1}{3}}\times (2^4)^{\frac{1}{3}}}{(2^5)^{-\frac{1}{3}}}$

Using properties of exponent, we get

$=\dfrac{2^{\frac{3}{3}}\times 2^{\frac{4}{3}}}{2^{-\frac{5}{3}}}$      $[\because (a^m)^n=a^{mn}]$

$=\dfrac{2^{1}\times 2^{\frac{4}{3}}}{2^{-\frac{5}{3}}}$

$=\dfrac{2^{1+\frac{4}{3}}}{2^{-\frac{5}{3}}}$      $[\because a^ma^n=a^{m+n}]$

$=\dfrac{2^{\frac{7}{3}}}{2^{-\frac{5}{3}}}$  

$=2^{\frac{7}{3}-(-\frac{5}{3})}$        $[\because \dfrac{a^m}{a^n}=a^{m-n}]$

$=2^{\frac{7}{3}+\frac{5}{3}}$  

$=2^{\frac{12}{3}}$  

$=2^4$  

$=16$  

Therefore, the value of given expression is 16.