Simplify $\frac{\sqrt{108}}{\sqrt{24}}$

Steps to Solve

1. Simplify $\sqrt{108}$

Find the prime factorization of 108: $108 = 2^2 \cdot 3^3 = 2^2 \cdot 3^2 \cdot 3$. Then, $\sqrt{108} = \sqrt{2^2 \cdot 3^2 \cdot 3} = \sqrt{2^2} \cdot \sqrt{3^2} \cdot \sqrt{3} = 2 \cdot 3 \cdot \sqrt{3} = 6\sqrt{3}$.

2. Simplify $\sqrt{24}$

Find the prime factorization of 24: $24 = 2^3 \cdot 3 = 2^2 \cdot 2 \cdot 3$. Then, $\sqrt{24} = \sqrt{2^2 \cdot 2 \cdot 3} = \sqrt{2^2} \cdot \sqrt{2 \cdot 3} = 2\sqrt{6}$.

3. Rewrite the fraction

Substitute the simplified square roots into the original fraction: $\frac{\sqrt{108}}{\sqrt{24}} = \frac{6\sqrt{3}}{2\sqrt{6}}$.

4. Simplify the fraction

Simplify the fraction by dividing the coefficients and the square roots separately: $\frac{6\sqrt{3}}{2\sqrt{6}} = \frac{6}{2} \cdot \frac{\sqrt{3}}{\sqrt{6}} = 3 \cdot \sqrt{\frac{3}{6}} = 3 \cdot \sqrt{\frac{1}{2}} = 3 \cdot \frac{1}{\sqrt{2}}$.

5. Rationalize the denominator

Multiply the numerator and the denominator by $\sqrt{2}$ to rationalize the denominator: $3 \cdot \frac{1}{\sqrt{2}} = 3 \cdot \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = 3 \cdot \frac{\sqrt{2}}{2} = \frac{3\sqrt{2}}{2}$.