Question
Simplify: $\sqrt{72 n^{7}}$

Provide your answer below:

Step 1 ：The given expression is \(\sqrt{72 n^{7}}\).

Step 2 ：Break down 72 and \(n^{7}\) into their prime factors. 72 can be factored into \(2^{3} * 3^{2}\) and \(n^{7}\) can be factored into \(n^{3} * n^{3} * n\).

Step 3 ：The square root of a number is found by pairing the factors and taking one from each pair.

Step 4 ：For 72, we can take one 2 from the pair of 2's and one 3 from the pair of 3's.

Step 5 ：For \(n^{7}\), we can take one n from each pair of n's. However, there will be one n left over that does not have a pair. This will remain under the square root.

Step 6 ：So, the simplified form of \(\sqrt{72 n^{7}}\) is \(6n^{3}\sqrt{n}\).

Step 7 ：Final Answer: The simplified form of \(\sqrt{72 n^{7}}\) is \(\boxed{6n^{3}\sqrt{2n}}\).