Question: Simplify. \(\sqrt{75v^4}\) Assume that the variable \(v\) represents a positive real number.

Simplify.

\(\sqrt{75v^4}\)

Assume that the variable \(v\) represents a positive real number.

Solution

To simplify the expression \(\sqrt{75v^4}\), follow these steps: First, break down the expression under the square root: \[ 75v^4 = 75 \cdot v^4 \] Next, factor 75 into its prime factors: \[ 75 = 5 \cdot 15 = 5 \cdot 3 \cdot 5 = 5^2 \cdot 3 \] Substitute this into the expression: \[ \sqrt{75v^4} = \sqrt{5^2 \cdot 3 \cdot v^4} \] Apply the property of square roots that allows us to separate the square root of a product into the product of square roots: \[ \sqrt{5^2 \cdot 3 \cdot v^4} = \sqrt{5^2} \cdot \sqrt{3} \cdot \sqrt{v^4} \] Simplify each square root individually: \[ \sqrt{5^2} = 5, \quad \sqrt{3} = \sqrt{3}, \quad \sqrt{v^4} = v^2 \] Combine these simplified parts: \[ 5v^2\sqrt{3} \] So, the simplified expression is \(5v^2\sqrt{3}\).