products of irrational square roots practice

Question

what is 3^√54 simplified
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Answer 1

To simplify \( 3^{\sqrt{54}} \), we can start with simplifying \(\sqrt{54}\).

First, we factor \(54\):

\[ 54 = 9 \times 6 = 3^2 \times 6 \]

Now we can simplify \(\sqrt{54}\):

\[ \sqrt{54} = \sqrt{9 \times 6} = \sqrt{9} \times \sqrt{6} = 3\sqrt{6} \]

Now we substitute this back into \(3^{\sqrt{54}}\):

\[ 3^{\sqrt{54}} = 3^{3\sqrt{6}} = (3^3)^{\sqrt{6}} = 27^{\sqrt{6}} \]

Thus, the simplified form of \( 3^{\sqrt{54}} \) is:

\[ 27^{\sqrt{6}} \]

So, the final answer is \( 27^{\sqrt{6}} \).