Solve frac{21sqrt{15}}{sqrt{12}+5sqrt{3}}

$$\frac { 21 \sqrt { 15 } } { \sqrt { 12 } + 5 \sqrt { 3 } }$$

Evaluate

$$3\sqrt{5}\approx 6.708203932$$

Solution Steps

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

$$\frac{21\sqrt{15}}{2\sqrt{3}+5\sqrt{3}}$$

Combine 2\sqrt{3} and 5\sqrt{3} to get 7\sqrt{3}.

$$\frac{21\sqrt{15}}{7\sqrt{3}}$$

Cancel out 7 in both numerator and denominator.

$$\frac{3\sqrt{15}}{\sqrt{3}}$$

Rationalize the denominator of \frac{3\sqrt{15}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{3\sqrt{15}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{3\sqrt{15}\sqrt{3}}{3}$$

Factor 15=3\times 5. Rewrite the square root of the product \sqrt{3\times 5} as the product of square roots \sqrt{3}\sqrt{5}.

$$\frac{3\sqrt{3}\sqrt{5}\sqrt{3}}{3}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{3\times 3\sqrt{5}}{3}$$

Cancel out 3 and 3.

$$3\sqrt{5}$$