Question

Which of the following is equal to sqrt 50a^6b^7

Answer 1

Your question implies there are choices given with possible answers. As I do not have those in front of me, I will show you how to simplify the radical. It is likely that the answer you seek is the simplified radical.

To simplify $\sqrt{50 a^{6} b^{7} }$ we use the following identity: for positive values of a and b, $\sqrt{ab}= \sqrt{a} \sqrt{b}$

That identity allows us to re-write what we are given as $\sqrt{50} \sqrt{ a^6 } \sqrt{b^{7}$. Let us now consider each of these separately.

Though 50 is not a perfect square it can be written as 25 x 2 and 25 is a perfect square. Thus, we obtain $\sqrt{50}= \sqrt{25} \sqrt{2}=5 \sqrt{2}$

Next we consider $a^{6}$ which is a perfect square as it is equal to $(a^{3}) (a^{3})$. Thus, $\sqrt{a^{6}} = a^{3}$

Lastly, we consider $b^{7}$ which can be written as $(b^{1})( b^{6})$ and since $b^{6}$ is a perfect square we obtain the following: $\sqrt{b ^{7} } =\sqrt{b ^{1}} \sqrt{b^6} = b^{3} \sqrt{b}$

Putting that all together we obtain $\sqrt{50 a^{6} b^{7} }=25a ^{3}b ^{3} \sqrt{2b}$