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

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Answer 1 · 2015-02-04T18:58:28

We'll need a few properties.

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.
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.
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
$\sqrt{75a^12b^3c^5} = 5a^6bc^2\sqrt{3bc}$

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