Problem 74 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

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

Step 1: Prime Factorization of 80

Perform the prime factorization of 80. It can be broken down into \(2 \times 2 \times 2 \times 2 \times 5\), or \(2^4 \times 5\).

Step 2: Simplification of the square root

Simplify \( \sqrt{80} \) to \( \sqrt{2^4 \times 5} \). The square root of \(2^4\) is 2 squared or 4, and 5 remains inside the square root since it has no pair. Therefore, \( \sqrt{80} \) is simplified to \(4\sqrt{5}\).

Step 3: Simplification of the entire expression

Finally, multiply \( \frac{1}{2} \) by \(4\sqrt{5}\) which results in \(2\sqrt{5}\).