Questions: Use rational exponents to simplify the following radical, then convert back to radical notation. Assume that all variables represent positive numbers. sqrt[4]16 x^2

Solution Steps

The given expression is: \[ \sqrt[4]{16 x^{2}} \] Using rational exponents, this can be rewritten as: \[ (16 x^{2})^{1/4} \]

First, simplify \(16^{1/4}\): \[ 16^{1/4} = (2^4)^{1/4} = 2^{4 \cdot \frac{1}{4}} = 2^1 = 2 \] Next, simplify \(x^{2 \cdot \frac{1}{4}}\): \[ x^{2 \cdot \frac{1}{4}} = x^{\frac{1}{2}} = \sqrt{x} \] Combining these results: \[ (16 x^{2})^{1/4} = 2 \cdot \sqrt{x} \]

The expression \(2 \cdot \sqrt{x}\) is already in radical notation, as \(\sqrt{x}\) is equivalent to \(x^{1/2}\).

Final Answer