Problem 67 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

The simplified form of \(\frac{1}{2} \sqrt{80}\) is \(2\sqrt{5}\).

Step 1: Identify the perfect squares that factor into 80

Perfect squares are numbers that are the product of a number with itself. The perfect squares less than 80 are 1, 4, 9, 16, 25, 36, 49, 64. The largest perfect square that factors into 80 is 16. So, we can rewrite \(\sqrt{80}\) as \(\sqrt{16*5}\).

Step 2: Simplify the square root

The square root of a product can be written as the product of the square roots. Hence, \(\sqrt{16*5}\) can be written as \(\sqrt{16} * \sqrt{5}\). Since \(\sqrt{16} = 4\), we can simplify \(\sqrt{16} * \sqrt{5}\) to \(4\sqrt{5}\).

Step 3: Multiply by the fraction in front of the radical

Now, multiply \(4\sqrt{5}\) by the fraction \(\frac{1}{2}\) in front of the radical symbol: \(\frac{1}{2} * 4\sqrt{5} = 2\sqrt{5}\).