Solve (5/sqrt{3/2+sqrt{3}})

$$( \frac { 5 } { \frac { \sqrt { 3 } } { 2 + \sqrt { 3 } } } )$$

Evaluate

$$\frac{10\sqrt{3}}{3}+5\approx 10.773502692$$

Solution Steps

Divide 5 by \frac{\sqrt{3}}{2+\sqrt{3}} by multiplying 5 by the reciprocal of \frac{\sqrt{3}}{2+\sqrt{3}}.

$$\frac{5\left(2+\sqrt{3}\right)}{\sqrt{3}}$$

Rationalize the denominator of \frac{5\left(2+\sqrt{3}\right)}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{5\left(2+\sqrt{3}\right)\sqrt{3}}{\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{5\left(2+\sqrt{3}\right)\sqrt{3}}{3}$$

Use the distributive property to multiply 5 by 2+\sqrt{3}.

$$\frac{\left(10+5\sqrt{3}\right)\sqrt{3}}{3}$$

Use the distributive property to multiply 10+5\sqrt{3} by \sqrt{3}.

$$\frac{10\sqrt{3}+5\left(\sqrt{3}\right)^{2}}{3}$$

The square of \sqrt{3} is 3.

$$\frac{10\sqrt{3}+5\times 3}{3}$$

Multiply 5 and 3 to get 15.

$$\frac{10\sqrt{3}+15}{3}$$