Question

Simplify the radical expression. $$ \frac{1}{2} \sqrt{80} $$

Short answer

The simplified form of the given radical expression is \(2\sqrt{5}\)

Step-by-step solution

Identify the terms

The first step is to identify the terms in the radical expression. Here, \(\frac{1}{2}\) is a constant term outside the radical and 80 under the square root.

Prime factorization

Use prime factorization to break down 80 into its base prime numbers. 80 can be written as \(2^4*5\).

Simplify radical

Using properties of radicals, you can factor out the perfect square from under the square root. Here, we have \(2^4\), which is a perfect square. This can be simplified to \(2*2 = 4\). So, the square root of 80 simplifies to \(4\sqrt{5}\).

Multiply with the constant

To find the final simplified form, multiply the constant term \(\frac{1}{2}\) with the simplified square root form \(4\sqrt{5}\).

Final Simplification

\(\frac{1}{2}\) multiplied by \(4\sqrt{5}\) gives \(2\sqrt{5}\), which is the simplified radical expression.