How do you determine which is greater ${(3^{5})}^{4}$ $(3^5)^4$ or $3^{10}$ $3^10$ ?

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Answer 1 · 2018-05-27T23:15:56

${(3^{5})}^{4} > 3^{10}$ $(3^5)^4>3^10$

We want to compare ${(3^{5})}^{4}$ $(3^5)^4$ with $3^{10}$ $3^10$ . To do so, we use the following exponent law:

${(a^{m})}^{n} = a^{m n}$ $(a^m)^n=a^(mn)$

So

${(3^{5})}^{4} = 3^{20}$ $(3^5)^4=3^20$

$20 > 10$ $20>10$ hence $3^{20} > 3^{10}$ $3^20>3^10$

We note that $3^{5^{4}} > {(3^{5})}^{4}$ $3^(5^4)>(3^5)^4$ because when evaluating exponent towers, we start from the top.

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