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I am aware of the following rule: $a^{\frac{b}{c}} = (a^b)^{\frac{1}{c}}$ AND $a^\frac{b}{c} = (a^\frac{1}{c})^b$

I have a problem as follows: $32^\frac{3}{5}$

I simplify it: $(32^\frac{1}{5})^3$

The problem is it took me some time to realize what $32^\frac{1}{5}$ is because raising something to the power of 5 is not too obvious what the solution could be. Ultimately, I realized it was $2^5$ but that came with much trial and error. Are there any techniques available to help me resolve these kinds of fractional exponents quicker?

Raven
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JohnMerlino
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  • You should use parentheses. a^b/c should be read as $(a^b)/c$, not $a^{(b/c)}$ – Ross Millikan Feb 23 '14 at 05:44
  • Break it down into a smaller number. Since $32$ is even start with dividing it by $2$. $$32=2\cdot16=2\cdot(2\cdot8)=2\cdot(2\cdot2^3)=2^5$$ – Zhoe Feb 23 '14 at 05:46

2 Answers2

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As $\displaystyle 32=2^5, 32^{\dfrac15}=(2^5)^{\dfrac15}=2^1$

$\displaystyle\implies32^{\dfrac35}=(32^{\dfrac15})^3=\cdots$

lab bhattacharjee
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It may be useful to memorize a few powers of $2$: $1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024$.

Robert Israel
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