Solve Nested Radicals: Cube Root of Square Root of 36

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)

00:14 The result equals the number (A) to the root of order (B times C)

00:18 We will apply this formula to our exercise

00:24 Calculate the order multiplication

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

00:36 The result equals the number (A) to the power of their quotient (B divided by C)

00:39 We will apply this formula to our exercise

00:43 This is the solution

Step-by-Step Solution

To solve this problem, let's analyze and simplify the given expression $\sqrt[3]{\sqrt{36}}$ .

Step 1: Identify the root operations. We have a square root, $\sqrt{36}$ , and a cube root, $\sqrt[3]{\ldots}$ .
Step 2: Use the formula for roots for a root of a root: $\sqrt[m]{\sqrt[n]{x}} = x^{\frac{1}{mn}}$ .
Step 3: Apply this formula to the problem. In this case, the first operation is a square root, which can be written as $36^{\frac{1}{2}}$ , and the second operation is a cube root. Therefore, $\sqrt[3]{\sqrt{36}} = (36^{\frac{1}{2}})^{\frac{1}{3}}$ .
Step 4: Simplify using the power of a power rule, which allows us to multiply exponents: $(36^{\frac{1}{2}})^{\frac{1}{3}} = 36^{\frac{1}{2 \times 3}} = 36^{\frac{1}{6}}$ .

Thus, the expression $\sqrt[3]{\sqrt{36}}$ simplifies to $36^{\frac{1}{6}}$ .