Question: Simplify. \[ \sqrt[4]{256w^4} Assume that the variable represents a positive real number.

Simplify. \[ \sqrt[4]{256w^4} Assume that the variable represents a positive real number.

Solution

To simplify the expression \(\sqrt[4]{256w^4}\), let’s follow these steps: First, recognize that we’re looking for the fourth root of the product of a number and a variable raised to a power. Separate the terms inside the radical: \[ \sqrt[4]{256 \cdot w^4} \] Apply the property of radicals: \(\sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b}\): \[ \sqrt[4]{256} \cdot \sqrt[4]{w^4} \] Simplify each part separately: Calculate \(\sqrt[4]{256}\): \[ 256 = 2^8 \] Therefore: \[ \sqrt[4]{256} = \sqrt[4]{2^8} = 2^{8/4} = 2^2 = 4 \] Now simplify \(\sqrt[4]{w^4}\): \[ \sqrt[4]{w^4} = w^{4/4} = w^1 = w \] Combine the simplified parts: \[ 4 \cdot w = 4w \] Therefore, the simplified expression is \(4w\).