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

Complete the following exercise:

$\sqrt[4]{\sqrt[3]{49\cdot x}}=$

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 $\sqrt[12]{49x}$.