Simplify the Nested Radical: Fifth Root of Square Root of x^20

Question

Complete the following exercise:

$\sqrt[5]{\sqrt{x^{20}}}=$

00:00 Solve the following problem

00:03 A "regular" root is of the order 2

00:09 When we have a number (A) raised to the power (B) with the root order (C)

00:19 The result equals the number (A) with the root order of their product (B times C)

00:24 We will apply this formula to our exercise

00:31 Let's calculate the multiplication of the orders

00:41 When we have a number (A) raised to the power (B) with the root order (C)

00:46 The result equals the number (A) raised to the power of their quotient (B divided by C)

00:49 We will apply this formula to our exercise

00:53 Proceed to calculate the division of powers

00:57 This is the solution

To solve the problem $\sqrt[5]{\sqrt{x^{20}}}$ , we will convert the roots into fractional exponents and simplify:

The expression $\sqrt{x^{20}}$ can be represented as:

$(x^{20})^{\frac{1}{2}} = x^{20 \cdot \frac{1}{2}} = x^{10}$

Now, taking the fifth root of $x^{10}$ , we have:

$(x^{10})^{\frac{1}{5}} = x^{10 \cdot \frac{1}{5}} = x^{2}$

Therefore, the original expression $\sqrt[5]{\sqrt{x^{20}}}$ simplifies to $x^2$ .

$x^2$