Question

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Answer 1

First we rewrite the expression:
 root (8x ^ 7y ^ 8)
 (8x ^ 7y ^ 8) ^ (1/2)
 Properties of exponents:
 (8 ^ (1/2) x ^ (7/2) y ^ (8/2)
 We rewrite:
 (8 ^ (1/2) x ^ (7/2) y ^ 4
 (2 * 2 ^ (1/2) x ^ 3y ^ 4x ^ (1/2)
 (2 * x ^ 3 * y ^ 4 * (2x) ^ (1/2)
 Answer:
 (2*x^3*y^4*(2x)^(1/2)
 option 3

Answer 2

To take out terms outside the radical we need to divide the power of the term by the index of the radical; the quotient will be the power of the term outside the radical, and the remainder will be the power of the term inside the radical. 
First, lets factor 8:
$8=2 ^{3}$
Now we can divide the power of the term, 3, by the index of the radical 2:
$\frac{3}{2}$ = 1 with a remainder of 1
Next, lets do the same for our second term $x^{7}$:
$\frac{7}{2}$ = 3 with a remainder of 1
Again, lets do the same for our third term $y^{8}$:
$\frac{8}{2} =4$ with no remainder, so this term will come out completely.

Finally, lets take our terms out of the radical:
$\sqrt{8x^{7} y^{8} }= \sqrt{ 2^{3} x^{7} y^{8} } =2 x^{3} y^{4} \sqrt{2x}$

We can conclude that the correct answer is the third one.