Solve frac{frac{7sqrt{3}}{8}+sqrt{3}}{frac{3sqrt{15}}{4}}

$$\frac { \frac { 7 \sqrt { 3 } } { 8 } + \sqrt { 3 } } { \frac { 3 \sqrt { 15 } } { 4 } }$$

Evaluate

$$\frac{\sqrt{5}}{2}\approx 1.118033989$$

Solution Steps

Combine \frac{7\sqrt{3}}{8} and \sqrt{3} to get \frac{15}{8}\sqrt{3}.

$$\frac{\frac{15}{8}\sqrt{3}}{\frac{3\sqrt{15}}{4}}$$

Divide \frac{15}{8}\sqrt{3} by \frac{3\sqrt{15}}{4} by multiplying \frac{15}{8}\sqrt{3} by the reciprocal of \frac{3\sqrt{15}}{4}.

$$\frac{\frac{15}{8}\sqrt{3}\times 4}{3\sqrt{15}}$$

Rationalize the denominator of \frac{\frac{15}{8}\sqrt{3}\times 4}{3\sqrt{15}} by multiplying numerator and denominator by \sqrt{15}.

$$\frac{\frac{15}{8}\sqrt{3}\times 4\sqrt{15}}{3\left(\sqrt{15}\right)^{2}}$$

The square of \sqrt{15} is 15.

$$\frac{\frac{15}{8}\sqrt{3}\times 4\sqrt{15}}{3\times 15}$$

Express \frac{15}{8}\times 4 as a single fraction.

$$\frac{\frac{15\times 4}{8}\sqrt{3}\sqrt{15}}{3\times 15}$$

Multiply 15 and 4 to get 60.

$$\frac{\frac{60}{8}\sqrt{3}\sqrt{15}}{3\times 15}$$

Reduce the fraction \frac{60}{8} to lowest terms by extracting and canceling out 4.

$$\frac{\frac{15}{2}\sqrt{3}\sqrt{15}}{3\times 15}$$

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}.

$$\frac{\frac{15}{2}\sqrt{3}\sqrt{3}\sqrt{5}}{3\times 15}$$

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

$$\frac{\frac{15}{2}\times 3\sqrt{5}}{3\times 15}$$

Multiply 3 and 15 to get 45.

$$\frac{\frac{15}{2}\times 3\sqrt{5}}{45}$$

Express \frac{15}{2}\times 3 as a single fraction.

$$\frac{\frac{15\times 3}{2}\sqrt{5}}{45}$$

Multiply 15 and 3 to get 45.

$$\frac{\frac{45}{2}\sqrt{5}}{45}$$

Divide \frac{45}{2}\sqrt{5} by 45 to get \frac{1}{2}\sqrt{5}.

$$\frac{1}{2}\sqrt{5}$$