Simplify the Nested Radical: Sixth Root of Square Root of x^12

Video Solution

Solution Steps

00:00 Solve the following problem

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

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

00:15 The result equals the number (A) to the root of order (B times C)

00:18 Let's apply this formula to our exercise

00:23 Calculate the multiplication of the orders

00:31 When we have a number (A) to the power of (B) in a root of order (C)

00:35 The result equals the number (A) to the power of (B divided by C)

00:40 Let's apply this formula to our exercise

00:43 Calculate the division of powers

00:47 This is the solution

Step-by-Step Solution

To solve $\sqrt[6]{\sqrt{x^{12}}}$ , we will follow these steps:

Step 1: Simplify the inner radical expression $\sqrt{x^{12}}$ .
Step 2: Use the property of roots, expressing $\sqrt{x^{12}}$ as a power of $x$ .
Step 3: Use the result from step 1 in the outer root expression.
Step 4: Simplify the entire expression using exponent rules.

Now, let's perform each of these steps:

Step 1: Simplify $\sqrt{x^{12}}$ .
$\sqrt{x^{12}} = x^{12/2} = x^6$ .

Step 2: Simplify the outer expression $\sqrt[6]{x^6}$ .
$\sqrt[6]{x^6} = (x^6)^{1/6}$ .

Step 3: Apply the exponent rule $(a^m)^{n} = a^{m \times n}$ .
$(x^6)^{1/6} = x^{6 \times 1/6} = x^1 = x$ .

Therefore, the simplified expression is $\boxed{x}$ .

Thus, the solution to $\sqrt[6]{\sqrt{x^{12}}}$ is $x$ .