Problem 77 - Algebra 1: Concepts and Skills

Question

Simplify the radical expression. $$ \frac{2}{3} \sqrt{300} $$

Step-by-Step Solution

Answer

\(\frac{20}{3}\sqrt{3}\)

Step 1: Break down 300 into prime factors

Start by breaking 300 down into its prime factors. The prime factorisation of 300 is \(2 * 2 * 3 * 5 * 5\). So, we can rewrite the square root of 300 as the square root of \(2 * 2 * 3 * 5 * 5\).

Step 2: Simplify the Square Root

Next, simplify the square root by taking out pairs of prime factors as a single number. There are two pairs of prime factors: \(2 * 2\) and \(5 * 5\). The square root of \(2 * 2\) is 2 and the square root of \(5 * 5\) is 5. We are left with the 3 under the square root because it does not have a pair. Therefore, \(\sqrt{300}\) simplifies to \(2 * 5\sqrt{3}\), or \(10\sqrt{3}\).

Step 3: Multiply by the Fraction

Finally, multiply the simplified square root \(10\sqrt{3}\) by the fraction \(\frac{2}{3}\) at the beginning of the expression. Multiplying \(\frac{2}{3}\) by 10 gives \(\frac{20}{3}\). Therefore, the solution is \(\frac{20}{3}\sqrt{3}\).