Solve 5/4-sqrt{11}+1/3+sqrt{7}-6/sqrt{7-2}-frac{sqrt{7}-5}{2}

$$\frac{ 5 }{ 4- \sqrt{ 11 } } + \frac{ 1 }{ 3+ \sqrt{ 7 } } - \frac{ 6 }{ \sqrt{ 7 } -2 } - \frac{ \sqrt{ 7 } -5 }{ 2 }$$

Evaluate

$$\sqrt{11}+4-3\sqrt{7}\approx -0.620629143$$

Short Solution Steps

Rationalize the denominator of \frac{5}{4-\sqrt{11}} by multiplying numerator and denominator by 4+\sqrt{11}.

$$\frac{5\left(4+\sqrt{11}\right)}{\left(4-\sqrt{11}\right)\left(4+\sqrt{11}\right)}+\frac{1}{3+\sqrt{7}}-\frac{6}{\sqrt{7}-2}-\frac{\sqrt{7}-5}{2}$$

Consider \left(4-\sqrt{11}\right)\left(4+\sqrt{11}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{5\left(4+\sqrt{11}\right)}{4^{2}-\left(\sqrt{11}\right)^{2}}+\frac{1}{3+\sqrt{7}}-\frac{6}{\sqrt{7}-2}-\frac{\sqrt{7}-5}{2}$$

Square 4. Square \sqrt{11}.

$$\frac{5\left(4+\sqrt{11}\right)}{16-11}+\frac{1}{3+\sqrt{7}}-\frac{6}{\sqrt{7}-2}-\frac{\sqrt{7}-5}{2}$$

Subtract 11 from 16 to get 5.

$$\frac{5\left(4+\sqrt{11}\right)}{5}+\frac{1}{3+\sqrt{7}}-\frac{6}{\sqrt{7}-2}-\frac{\sqrt{7}-5}{2}$$

Cancel out 5 and 5.

$$4+\sqrt{11}+\frac{1}{3+\sqrt{7}}-\frac{6}{\sqrt{7}-2}-\frac{\sqrt{7}-5}{2}$$

Rationalize the denominator of \frac{1}{3+\sqrt{7}} by multiplying numerator and denominator by 3-\sqrt{7}.

$$4+\sqrt{11}+\frac{3-\sqrt{7}}{\left(3+\sqrt{7}\right)\left(3-\sqrt{7}\right)}-\frac{6}{\sqrt{7}-2}-\frac{\sqrt{7}-5}{2}$$

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}.

$$4+\sqrt{11}+\frac{3-\sqrt{7}}{3^{2}-\left(\sqrt{7}\right)^{2}}-\frac{6}{\sqrt{7}-2}-\frac{\sqrt{7}-5}{2}$$

Square 3. Square \sqrt{7}.

$$4+\sqrt{11}+\frac{3-\sqrt{7}}{9-7}-\frac{6}{\sqrt{7}-2}-\frac{\sqrt{7}-5}{2}$$

Subtract 7 from 9 to get 2.

$$4+\sqrt{11}+\frac{3-\sqrt{7}}{2}-\frac{6}{\sqrt{7}-2}-\frac{\sqrt{7}-5}{2}$$

Rationalize the denominator of \frac{6}{\sqrt{7}-2} by multiplying numerator and denominator by \sqrt{7}+2.

$$4+\sqrt{11}+\frac{3-\sqrt{7}}{2}-\frac{6\left(\sqrt{7}+2\right)}{\left(\sqrt{7}-2\right)\left(\sqrt{7}+2\right)}-\frac{\sqrt{7}-5}{2}$$

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

$$4+\sqrt{11}+\frac{3-\sqrt{7}}{2}-\frac{6\left(\sqrt{7}+2\right)}{\left(\sqrt{7}\right)^{2}-2^{2}}-\frac{\sqrt{7}-5}{2}$$

Square \sqrt{7}. Square 2.

$$4+\sqrt{11}+\frac{3-\sqrt{7}}{2}-\frac{6\left(\sqrt{7}+2\right)}{7-4}-\frac{\sqrt{7}-5}{2}$$

Subtract 4 from 7 to get 3.

$$4+\sqrt{11}+\frac{3-\sqrt{7}}{2}-\frac{6\left(\sqrt{7}+2\right)}{3}-\frac{\sqrt{7}-5}{2}$$

Divide 6\left(\sqrt{7}+2\right) by 3 to get 2\left(\sqrt{7}+2\right).

$$4+\sqrt{11}+\frac{3-\sqrt{7}}{2}-2\left(\sqrt{7}+2\right)-\frac{\sqrt{7}-5}{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 4+\sqrt{11} times \frac{2}{2}.

$$\frac{2\left(4+\sqrt{11}\right)}{2}+\frac{3-\sqrt{7}}{2}-2\left(\sqrt{7}+2\right)-\frac{\sqrt{7}-5}{2}$$

Since \frac{2\left(4+\sqrt{11}\right)}{2} and \frac{3-\sqrt{7}}{2} have the same denominator, add them by adding their numerators.

$$\frac{2\left(4+\sqrt{11}\right)+3-\sqrt{7}}{2}-2\left(\sqrt{7}+2\right)-\frac{\sqrt{7}-5}{2}$$

Do the multiplications in 2\left(4+\sqrt{11}\right)+3-\sqrt{7}.

$$\frac{8+2\sqrt{11}+3-\sqrt{7}}{2}-2\left(\sqrt{7}+2\right)-\frac{\sqrt{7}-5}{2}$$

Do the calculations in 8+2\sqrt{11}+3-\sqrt{7}.

$$\frac{11+2\sqrt{11}-\sqrt{7}}{2}-2\left(\sqrt{7}+2\right)-\frac{\sqrt{7}-5}{2}$$

Use the distributive property to multiply 2 by \sqrt{7}+2.

$$4+\sqrt{11}+\frac{3-\sqrt{7}}{2}-\left(2\sqrt{7}+4\right)-\frac{\sqrt{7}-5}{2}$$

To find the opposite of 2\sqrt{7}+4, find the opposite of each term.

$$4+\sqrt{11}+\frac{3-\sqrt{7}}{2}-2\sqrt{7}-4-\frac{\sqrt{7}-5}{2}$$

Subtract 4 from 4 to get 0.

$$\sqrt{11}+\frac{3-\sqrt{7}}{2}-2\sqrt{7}-\frac{\sqrt{7}-5}{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply \sqrt{11}-2\sqrt{7} times \frac{2}{2}.

$$\frac{2\left(\sqrt{11}-2\sqrt{7}\right)}{2}+\frac{3-\sqrt{7}}{2}-\frac{\sqrt{7}-5}{2}$$

Since \frac{2\left(\sqrt{11}-2\sqrt{7}\right)}{2} and \frac{3-\sqrt{7}}{2} have the same denominator, add them by adding their numerators.

$$\frac{2\left(\sqrt{11}-2\sqrt{7}\right)+3-\sqrt{7}}{2}-\frac{\sqrt{7}-5}{2}$$

Do the multiplications in 2\left(\sqrt{11}-2\sqrt{7}\right)+3-\sqrt{7}.

$$\frac{2\sqrt{11}-4\sqrt{7}+3-\sqrt{7}}{2}-\frac{\sqrt{7}-5}{2}$$

Do the calculations in 2\sqrt{11}-4\sqrt{7}+3-\sqrt{7}.

$$\frac{2\sqrt{11}-5\sqrt{7}+3}{2}-\frac{\sqrt{7}-5}{2}$$

Since \frac{2\sqrt{11}-5\sqrt{7}+3}{2} and \frac{\sqrt{7}-5}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\sqrt{11}-5\sqrt{7}+3-\left(\sqrt{7}-5\right)}{2}$$

Do the multiplications in 2\sqrt{11}-5\sqrt{7}+3-\left(\sqrt{7}-5\right).

$$\frac{2\sqrt{11}-5\sqrt{7}+3-\sqrt{7}+5}{2}$$

Do the calculations in 2\sqrt{11}-5\sqrt{7}+3-\sqrt{7}+5.

$$\frac{2\sqrt{11}-6\sqrt{7}+8}{2}$$

Divide each term of 2\sqrt{11}-6\sqrt{7}+8 by 2 to get \sqrt{11}-3\sqrt{7}+4.

$$\sqrt{11}-3\sqrt{7}+4$$

Factor

$$\sqrt{11} + 4 - 3 \sqrt{7} = -0.620629143$$