Solve Nested Radicals: Fourth Root of Cube Root with 49x

Video Solution

Solution Steps

00:00 Solve the following problem

00:03 When we have a number (A) to the power of (B) in a root of order (C)

00:07 The result equals the number (A) in a root of order of their product (B times C)

00:10 Apply this formula to our exercise

00:16 Calculate the product of the order

00:21 This is the solution

Step-by-Step Solution

To simplify the given expression $\sqrt[4]{\sqrt[3]{49 \cdot x}}$ , we will use the property of roots as fractional exponents.

Step 1: Convert the cube root to a fractional exponent. We have $\sqrt[3]{49 \cdot x} = (49 \cdot x)^{1/3}$ .
Step 2: Apply the fourth root to the result, expressed as a fractional exponent. Thus, $\sqrt[4]{(49 \cdot x)^{1/3}} = ((49 \cdot x)^{1/3})^{1/4}$ .
Step 3: Combine the exponents by multiplying the fractions: $((49 \cdot x)^{1/3})^{1/4} = (49 \cdot x)^{(1/3) \times (1/4)} = (49 \cdot x)^{1/12}$ .
Step 4: Recognize that the result $(49 \cdot x)^{1/12}$ can be expressed as $\sqrt[12]{49 \cdot x}$ .

Therefore, the solution to the problem is $\sqrt[12]{49x}$ .