Roots and powers - simplify

Algebra Level 2

I have got this one on my test a month ago. I liked it, so I wanted to share it.

Simplify this expression.

( a 2 ) 3 × a 2 3 ( a 1 3 ) 4 × a 5 \dfrac{(a^{2})^{3} \times \sqrt[3]{a^{2}}}{(a^{\frac{1}{3}})^{4} \times a^{5}}

a 3 \sqrt[3]{a} 1 a \sqrt{\frac{1}{a}} a 2 4 \sqrt[4]{a^{2}} a 6 a^{6} 1

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Dawid Szefler by Dawid Szefler , 21, Norway

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1 solution

Dawid Szefler
Jun 4, 2018

We will be using some of the exponent rules .

( a 2 ) 3 × a 2 3 ( a 1 3 ) 4 × a 5 \frac{(a^{2})^{3} \times \sqrt[3]{a^{2}}}{(a^{\frac{1}{3}})^{4} \times a^{5}} = a 6 × a 2 3 a 4 3 × a 5 \frac{a^6 \times a^\frac{2}{3}}{a^\frac{4}{3} \times a^5} = a 6 + 2 3 a 4 3 + 5 \frac{a^{6+\frac{2}{3}}}{a^{\frac{4}{3}+5}} = a 18 3 + 2 3 a 4 3 + 15 3 \frac{a^{\frac{18}{3}+\frac{2}{3}}}{a^{\frac{4}{3}+\frac{15}{3}}} = a 20 3 a 19 3 \frac{a^{\frac{20}{3}}}{a^{\frac{19}{3}}} = a 20 3 − 19 3 a^{\frac{20}{3}-\frac{19}{3}} = a 1 3 a^\frac{1}{3} = a 3 \sqrt[3]{a}

Here is the answer.

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