Multiplying exponents by fractional exponents and whole numbers to the power of fractional exponents

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I am trying to understand the different rules for multiplying exponents by fractional exponents and raising whole numbers by the power of fractional exponents. I have an idea but I'm trying to assure myself. Take the following example:

$(64x^4)^\frac{1}{3}$

Do I first find the cube root of 64 and bring it to the power of 1, which is 4? And then for the exponent of $x$, multiply it by 1 and divide it by 3, which is $\frac{4}{3}$ - resulting in:

$4x^\frac{4}{3}$

I am just trying to make sure I've got this down because I'm having a test next week. Thanks!

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1 Answer

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You are interested in $(64x^4)^{\frac{1}{3}}$, which can be decomposed as $64^{\frac{1}{3}}(x^4)^{\frac{1}{3}}$. Then you have dealt with $64^{\frac{1}{3}}=4$ correctly. For $(x^4)^{\frac{1}{3}}$, the law of exponents $(a^b)^c=a^{bc}$ gives $(x^4)^{\frac{1}{3}}=x^{\frac{4}{3}}$, giving a final result $(64x^4)^{\frac{1}{3}}=4x^{\frac{4}{3}}$ It looks like you misread the $3$ as a $2$ along the way.

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