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,
`√(ab)= √(a) √(b)`

That identity allows us to re-write what we are given as
`√(50) √( 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
`√(50)= √(25) √(2)=5 √(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)} √(b^6) = b^(3) √(b)`

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