How do you simplify ${(2^{\frac{1}{6}} \cdot 2^{\frac{1}{3}})}^{\frac{3}{2}}$ $(2^(1/6) * 2^(1/3)) ^(3/2)$ ?

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How do you simplify ${(2^{\frac{1}{6}} \cdot 2^{\frac{1}{3}})}^{\frac{3}{2}}$ $(2^(1/6) * 2^(1/3)) ^(3/2)$ ?

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Answer 1 · 2015-06-30T13:38:31

I found: $\sqrt[4]{8}$ $root4(8)$

Explanation:

You can use the following facts:
$x^{a} \cdot x^{b} = x^{a + b}$ $x^a*x^b=x^(a+b)$
${(x^{a})}^{b} = x^{a \cdot b}$ $(x^a)^b=x^(a*b)$
So you get:
${(2^{\frac{1}{6} + \frac{1}{3}})}^{\frac{3}{2}} = {(2^{\frac{3}{6}})}^{\frac{3}{2}} = {(2^{\frac{1}{2}})}^{\frac{3}{2}} =$ $(2^(1/6+1/3))^(3/2)=(2^(3/6))^(3/2)=(2^(1/2))^(3/2)=$
$= 2^{\frac{1}{2} \cdot \frac{3}{2}} = 2^{\frac{3}{4}} = \sqrt[4]{2^{3}} = \sqrt[4]{8}$ $=2^(1/2*3/2)=2^(3/4)=root4(2^3)=root4(8)$

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How do you simplify ${(2^{\frac{1}{6}} \cdot 2^{\frac{1}{3}})}^{\frac{3}{2}}$ $(2^(1/6) * 2^(1/3)) ^(3/2)$ ?

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