Solve (9/5-2sqrt{3}*frac{5+2sqrt{3}}{5+2sqrt{3}}

$$( \frac { 9 } { 5 - 2 \sqrt { 3 } } \times \frac { 5 + 2 \sqrt { 3 } } { 5 + 2 \sqrt { 3 } }$$

Evaluate

$$\frac{18\sqrt{3}+45}{13}\approx 5.859762657$$

Solution Steps

Divide 5+2\sqrt{3} by 5+2\sqrt{3} to get 1.

$$\frac{9}{5-2\sqrt{3}}\times 1$$

Rationalize the denominator of \frac{9}{5-2\sqrt{3}} by multiplying numerator and denominator by 5+2\sqrt{3}.

$$\frac{9\left(5+2\sqrt{3}\right)}{\left(5-2\sqrt{3}\right)\left(5+2\sqrt{3}\right)}\times 1$$

Consider \left(5-2\sqrt{3}\right)\left(5+2\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{9\left(5+2\sqrt{3}\right)}{5^{2}-\left(-2\sqrt{3}\right)^{2}}\times 1$$

Calculate 5 to the power of 2 and get 25.

$$\frac{9\left(5+2\sqrt{3}\right)}{25-\left(-2\sqrt{3}\right)^{2}}\times 1$$

Expand \left(-2\sqrt{3}\right)^{2}.

$$\frac{9\left(5+2\sqrt{3}\right)}{25-\left(-2\right)^{2}\left(\sqrt{3}\right)^{2}}\times 1$$

Calculate -2 to the power of 2 and get 4.

$$\frac{9\left(5+2\sqrt{3}\right)}{25-4\left(\sqrt{3}\right)^{2}}\times 1$$

The square of \sqrt{3} is 3.

$$\frac{9\left(5+2\sqrt{3}\right)}{25-4\times 3}\times 1$$

Multiply 4 and 3 to get 12.

$$\frac{9\left(5+2\sqrt{3}\right)}{25-12}\times 1$$

Subtract 12 from 25 to get 13.

$$\frac{9\left(5+2\sqrt{3}\right)}{13}\times 1$$

Express \frac{9\left(5+2\sqrt{3}\right)}{13}\times 1 as a single fraction.

$$\frac{9\left(5+2\sqrt{3}\right)}{13}$$

Use the distributive property to multiply 9 by 5+2\sqrt{3}.

$$\frac{45+18\sqrt{3}}{13}$$