Solve {l}{frac{3+sqrt{7}}{3-sqrt{7}}}{-frac{3-sqrt{7}}{3+2sqrt{7}}}

$$\left. \begin{array} { l } { \frac { 3 + \sqrt { 7 } } { 3 - \sqrt { 7 } } } \\ { - \frac { 3 - \sqrt { 7 } } { 3 + 2 \sqrt { 7 } } } \end{array} \right.$$

Evaluate

$$\frac{48\sqrt{7}+175}{19}\approx 15.894529628$$

Short Solution Steps

Rationalize the denominator of \frac{3+\sqrt{7}}{3-\sqrt{7}} by multiplying numerator and denominator by 3+\sqrt{7}.

$$\frac{\left(3+\sqrt{7}\right)\left(3+\sqrt{7}\right)}{\left(3-\sqrt{7}\right)\left(3+\sqrt{7}\right)}-\frac{3-\sqrt{7}}{3+2\sqrt{7}}$$

Consider \left(3-\sqrt{7}\right)\left(3+\sqrt{7}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(3+\sqrt{7}\right)\left(3+\sqrt{7}\right)}{3^{2}-\left(\sqrt{7}\right)^{2}}-\frac{3-\sqrt{7}}{3+2\sqrt{7}}$$

Square 3. Square \sqrt{7}.

$$\frac{\left(3+\sqrt{7}\right)\left(3+\sqrt{7}\right)}{9-7}-\frac{3-\sqrt{7}}{3+2\sqrt{7}}$$

Subtract 7 from 9 to get 2.

$$\frac{\left(3+\sqrt{7}\right)\left(3+\sqrt{7}\right)}{2}-\frac{3-\sqrt{7}}{3+2\sqrt{7}}$$

Multiply 3+\sqrt{7} and 3+\sqrt{7} to get \left(3+\sqrt{7}\right)^{2}.

$$\frac{\left(3+\sqrt{7}\right)^{2}}{2}-\frac{3-\sqrt{7}}{3+2\sqrt{7}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3+\sqrt{7}\right)^{2}.

$$\frac{9+6\sqrt{7}+\left(\sqrt{7}\right)^{2}}{2}-\frac{3-\sqrt{7}}{3+2\sqrt{7}}$$

The square of \sqrt{7} is 7.

$$\frac{9+6\sqrt{7}+7}{2}-\frac{3-\sqrt{7}}{3+2\sqrt{7}}$$

Add 9 and 7 to get 16.

$$\frac{16+6\sqrt{7}}{2}-\frac{3-\sqrt{7}}{3+2\sqrt{7}}$$

Divide each term of 16+6\sqrt{7} by 2 to get 8+3\sqrt{7}.

$$8+3\sqrt{7}-\frac{3-\sqrt{7}}{3+2\sqrt{7}}$$

Rationalize the denominator of \frac{3-\sqrt{7}}{3+2\sqrt{7}} by multiplying numerator and denominator by 3-2\sqrt{7}.

$$8+3\sqrt{7}-\frac{\left(3-\sqrt{7}\right)\left(3-2\sqrt{7}\right)}{\left(3+2\sqrt{7}\right)\left(3-2\sqrt{7}\right)}$$

Consider \left(3+2\sqrt{7}\right)\left(3-2\sqrt{7}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$8+3\sqrt{7}-\frac{\left(3-\sqrt{7}\right)\left(3-2\sqrt{7}\right)}{3^{2}-\left(2\sqrt{7}\right)^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$8+3\sqrt{7}-\frac{\left(3-\sqrt{7}\right)\left(3-2\sqrt{7}\right)}{9-\left(2\sqrt{7}\right)^{2}}$$

Expand \left(2\sqrt{7}\right)^{2}.

$$8+3\sqrt{7}-\frac{\left(3-\sqrt{7}\right)\left(3-2\sqrt{7}\right)}{9-2^{2}\left(\sqrt{7}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$8+3\sqrt{7}-\frac{\left(3-\sqrt{7}\right)\left(3-2\sqrt{7}\right)}{9-4\left(\sqrt{7}\right)^{2}}$$

The square of \sqrt{7} is 7.

$$8+3\sqrt{7}-\frac{\left(3-\sqrt{7}\right)\left(3-2\sqrt{7}\right)}{9-4\times 7}$$

Multiply 4 and 7 to get 28.

$$8+3\sqrt{7}-\frac{\left(3-\sqrt{7}\right)\left(3-2\sqrt{7}\right)}{9-28}$$

Subtract 28 from 9 to get -19.

$$8+3\sqrt{7}-\frac{\left(3-\sqrt{7}\right)\left(3-2\sqrt{7}\right)}{-19}$$

Apply the distributive property by multiplying each term of 3-\sqrt{7} by each term of 3-2\sqrt{7}.

$$8+3\sqrt{7}-\frac{9-6\sqrt{7}-3\sqrt{7}+2\left(\sqrt{7}\right)^{2}}{-19}$$

Combine -6\sqrt{7} and -3\sqrt{7} to get -9\sqrt{7}.

$$8+3\sqrt{7}-\frac{9-9\sqrt{7}+2\left(\sqrt{7}\right)^{2}}{-19}$$

The square of \sqrt{7} is 7.

$$8+3\sqrt{7}-\frac{9-9\sqrt{7}+2\times 7}{-19}$$

Multiply 2 and 7 to get 14.

$$8+3\sqrt{7}-\frac{9-9\sqrt{7}+14}{-19}$$

Add 9 and 14 to get 23.

$$8+3\sqrt{7}-\frac{23-9\sqrt{7}}{-19}$$

Multiply both numerator and denominator by -1.

$$8+3\sqrt{7}-\frac{-23+9\sqrt{7}}{19}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 8+3\sqrt{7} times \frac{19}{19}.

$$\frac{19\left(8+3\sqrt{7}\right)}{19}-\frac{-23+9\sqrt{7}}{19}$$

Since \frac{19\left(8+3\sqrt{7}\right)}{19} and \frac{-23+9\sqrt{7}}{19} have the same denominator, subtract them by subtracting their numerators.

$$\frac{19\left(8+3\sqrt{7}\right)-\left(-23+9\sqrt{7}\right)}{19}$$

Do the multiplications in 19\left(8+3\sqrt{7}\right)-\left(-23+9\sqrt{7}\right).

$$\frac{152+57\sqrt{7}+23-9\sqrt{7}}{19}$$

Do the calculations in 152+57\sqrt{7}+23-9\sqrt{7}.

$$\frac{175+48\sqrt{7}}{19}$$

Factor

$$\frac{48 \sqrt{7} + 175}{19} = 15.89452962795265$$