Question

Using complete sentences, explain the two ways to simplify sqrt(180)
and provide the simplified form.

Answer 1

Oh, where, oh where is my prime factorization? I findeth it beneath the radical. What joy to see factors of 2. Pull them out until you are through.

180
2*90
2*2*45

Alas, 2 is done, perhaps 3 will give some?

2*2*45
2*2*3*15
2*2*3*3*5

No more threes, but what have we left? Only a 5 for the radical to heft.

\(\sqrt{2*2*3*3*5} = \sqrt{2\cdot 2}\cdot\sqrt{3\cdot 3}\cdot \sqrt{5}\)

"Is there more," we ask? Certainly, a simplify will complete the task.

\(\sqrt{2\cdot 2}\cdot\sqrt{3\cdot 3}\cdot \sqrt{5} = 2\cdot 3\cdot\sqrt{5}\)

Yet, only one thing we lack, then delight we return to the question's author.

\(2\cdot 3\cdot\sqrt{5} = 6\sqrt{5}\)

It just says "complete sentences". It doesn't say you have to be boring and technical. Branch out and have a little fun with it!