Question: Simplify. √8w^6 Assume that the variable represents a positive real number.

Simplify. √8w^6 Assume that the variable represents a positive real number.

Solution

To simplify the expression \(\sqrt[3]{8w^6}\): First, express the cube root as an exponent: \[ \sqrt[3]{8w^6} = (8w^6)^{\frac{1}{3}} \] Next, apply the exponent to both the 8 and \(w^6\): \[ = 8^{\frac{1}{3}} \cdot (w^6)^{\frac{1}{3}} \] Calculate \(8^{\frac{1}{3}}\): \[ 8^{\frac{1}{3}} = 2 \] Calculate \((w^6)^{\frac{1}{3}}\) by multiplying the exponents: \[ (w^6)^{\frac{1}{3}} = w^{6 \cdot \frac{1}{3}} = w^2 \] Combine the simplified terms: \[ 2 \cdot w^2 = 2w^2 \] The simplified expression is \(2w^2\).