Solve 1/11-2sqrt{30}-1/11+2sqrt{30}

$$\frac { 1 } { 11 - 2 \sqrt { 30 } } - \frac { 1 } { 11 + 2 \sqrt { 30 } }$$

Evaluate

$$4\sqrt{30}\approx 21.9089023$$

Solution Steps

Rationalize the denominator of \frac{1}{11-2\sqrt{30}} by multiplying numerator and denominator by 11+2\sqrt{30}.

$$\frac{11+2\sqrt{30}}{\left(11-2\sqrt{30}\right)\left(11+2\sqrt{30}\right)}-\frac{1}{11+2\sqrt{30}}$$

Consider \left(11-2\sqrt{30}\right)\left(11+2\sqrt{30}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{11+2\sqrt{30}}{11^{2}-\left(-2\sqrt{30}\right)^{2}}-\frac{1}{11+2\sqrt{30}}$$

Calculate 11 to the power of 2 and get 121.

$$\frac{11+2\sqrt{30}}{121-\left(-2\sqrt{30}\right)^{2}}-\frac{1}{11+2\sqrt{30}}$$

Expand \left(-2\sqrt{30}\right)^{2}.

$$\frac{11+2\sqrt{30}}{121-\left(-2\right)^{2}\left(\sqrt{30}\right)^{2}}-\frac{1}{11+2\sqrt{30}}$$

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

$$\frac{11+2\sqrt{30}}{121-4\left(\sqrt{30}\right)^{2}}-\frac{1}{11+2\sqrt{30}}$$

The square of \sqrt{30} is 30.

$$\frac{11+2\sqrt{30}}{121-4\times 30}-\frac{1}{11+2\sqrt{30}}$$

Multiply 4 and 30 to get 120.

$$\frac{11+2\sqrt{30}}{121-120}-\frac{1}{11+2\sqrt{30}}$$

Subtract 120 from 121 to get 1.

$$\frac{11+2\sqrt{30}}{1}-\frac{1}{11+2\sqrt{30}}$$

Anything divided by one gives itself.

$$11+2\sqrt{30}-\frac{1}{11+2\sqrt{30}}$$

Rationalize the denominator of \frac{1}{11+2\sqrt{30}} by multiplying numerator and denominator by 11-2\sqrt{30}.

$$11+2\sqrt{30}-\frac{11-2\sqrt{30}}{\left(11+2\sqrt{30}\right)\left(11-2\sqrt{30}\right)}$$

Consider \left(11+2\sqrt{30}\right)\left(11-2\sqrt{30}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$11+2\sqrt{30}-\frac{11-2\sqrt{30}}{11^{2}-\left(2\sqrt{30}\right)^{2}}$$

Calculate 11 to the power of 2 and get 121.

$$11+2\sqrt{30}-\frac{11-2\sqrt{30}}{121-\left(2\sqrt{30}\right)^{2}}$$

Expand \left(2\sqrt{30}\right)^{2}.

$$11+2\sqrt{30}-\frac{11-2\sqrt{30}}{121-2^{2}\left(\sqrt{30}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$11+2\sqrt{30}-\frac{11-2\sqrt{30}}{121-4\left(\sqrt{30}\right)^{2}}$$

The square of \sqrt{30} is 30.

$$11+2\sqrt{30}-\frac{11-2\sqrt{30}}{121-4\times 30}$$

Multiply 4 and 30 to get 120.

$$11+2\sqrt{30}-\frac{11-2\sqrt{30}}{121-120}$$

Subtract 120 from 121 to get 1.

$$11+2\sqrt{30}-\frac{11-2\sqrt{30}}{1}$$

Anything divided by one gives itself.

$$11+2\sqrt{30}-\left(11-2\sqrt{30}\right)$$

To find the opposite of 11-2\sqrt{30}, find the opposite of each term.

$$11+2\sqrt{30}-11-\left(-2\sqrt{30}\right)$$

The opposite of -2\sqrt{30} is 2\sqrt{30}.

$$11+2\sqrt{30}-11+2\sqrt{30}$$

Subtract 11 from 11 to get 0.

$$2\sqrt{30}+2\sqrt{30}$$

Combine 2\sqrt{30} and 2\sqrt{30} to get 4\sqrt{30}.

$$4\sqrt{30}$$