Question

How do you simplify `sqrt(75a^12b^3c^5)`?

Answer 1

We'll need a few properties.

1. First of all, let's recall that `\sqrt{a\cdot b}=\sqrt{a} \sqrt{b}`. This of course applies also to products of more than two factors.
2. I hope you will be ok with the fact that the square root of a number is that number to the power of `1/2`. If not, tell me in the comments and I will explain this exercise in another way.
3. The third properties is that `a^{b+c}=a^b\cdot a^c`.

If this things are ok, for the first point we can separate the roots:
`\sqrt{75a^12b^3c^5}=\sqrt{75}\sqrt{a^12}\sqrt{b^3}\sqrt{c^5}`

Factoring `75` in prime numbers, we have `\sqrt{75}=\sqrt{3\cdot 5^2}=\sqrt{3}\sqrt{5^2}=5\sqrt{3}`.

For `\sqrt{a^12}`, we have that `\sqrt{a^12}=a^{12/2}=a^6`.

For `\sqrt{b^3}`, we have that `\sqrt{b^3}=b^{3/2}=b^{1+{1}/{2}}=b\sqrt{b}`

For `\sqrt{c^5}`, we have that `\sqrt{c^5}=c^{5/2}=c^{2+{1}/{2}}=c^2\sqrt{c}`.

Putting all the pieces together, we have that
`\sqrt{75a^12b^3c^5} = 5a^6bc^2\sqrt{3bc}`