Question

Which expression is equivalent to

Answer 1

The answer for the question shown above is the first option: x^2(x^2)^1/4
 When you simplify the expression, you obtain the equivalent expression:

 (x^10)^1/4
 [(x^8)(x2)]^1/4
 x^2(x^2)^1/4

 Therefore, the asnwer is the option mentioned before.

Answer 2

Answer: the first option:

$x^2 (\sqrt[4]{x^2} )$

Explanation:

1) Since the exponent of the radicand (the letter inside the root) is less than the index of the radical you can extract partially the letter from the radical.

2) You divide the exponent (10) by the index of the radical (4) and that gives a quotient of 2 and a remainder of 2.

3) The quotient means that the letter appears out of the radicand with that exponent. That is x².

4) The remainder gives the exponent of the letter that stands inside the root.  That is x².

Therefore, the result is that x² leaves the root and x² stands inside the root.

You can also do it in this way:

$\sqrt[4]{x^{10}} = \sqrt[4]{x^8x^2} = \sqrt[4]{x^8} \sqrt[4]{x^2} = x^{2} \sqrt[4]{x^2}$