Solve 1/8-2sqrt{15}+1/8+2sqrt{15}

$$\frac{ 1 }{ 8-2 \sqrt{ 15 } } + \frac{ 1 }{ 8+2 \sqrt{ 15 } }$$

Evaluate

$$4$$

Short Solution Steps

Rationalize the denominator of \frac{1}{8-2\sqrt{15}} by multiplying numerator and denominator by 8+2\sqrt{15}.

$$\frac{8+2\sqrt{15}}{\left(8-2\sqrt{15}\right)\left(8+2\sqrt{15}\right)}+\frac{1}{8+2\sqrt{15}}$$

Consider \left(8-2\sqrt{15}\right)\left(8+2\sqrt{15}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{8+2\sqrt{15}}{8^{2}-\left(-2\sqrt{15}\right)^{2}}+\frac{1}{8+2\sqrt{15}}$$

Calculate 8 to the power of 2 and get 64.

$$\frac{8+2\sqrt{15}}{64-\left(-2\sqrt{15}\right)^{2}}+\frac{1}{8+2\sqrt{15}}$$

Expand \left(-2\sqrt{15}\right)^{2}.

$$\frac{8+2\sqrt{15}}{64-\left(-2\right)^{2}\left(\sqrt{15}\right)^{2}}+\frac{1}{8+2\sqrt{15}}$$

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

$$\frac{8+2\sqrt{15}}{64-4\left(\sqrt{15}\right)^{2}}+\frac{1}{8+2\sqrt{15}}$$

The square of \sqrt{15} is 15.

$$\frac{8+2\sqrt{15}}{64-4\times 15}+\frac{1}{8+2\sqrt{15}}$$

Multiply 4 and 15 to get 60.

$$\frac{8+2\sqrt{15}}{64-60}+\frac{1}{8+2\sqrt{15}}$$

Subtract 60 from 64 to get 4.

$$\frac{8+2\sqrt{15}}{4}+\frac{1}{8+2\sqrt{15}}$$

Rationalize the denominator of \frac{1}{8+2\sqrt{15}} by multiplying numerator and denominator by 8-2\sqrt{15}.

$$\frac{8+2\sqrt{15}}{4}+\frac{8-2\sqrt{15}}{\left(8+2\sqrt{15}\right)\left(8-2\sqrt{15}\right)}$$

Consider \left(8+2\sqrt{15}\right)\left(8-2\sqrt{15}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{8+2\sqrt{15}}{4}+\frac{8-2\sqrt{15}}{8^{2}-\left(2\sqrt{15}\right)^{2}}$$

Calculate 8 to the power of 2 and get 64.

$$\frac{8+2\sqrt{15}}{4}+\frac{8-2\sqrt{15}}{64-\left(2\sqrt{15}\right)^{2}}$$

Expand \left(2\sqrt{15}\right)^{2}.

$$\frac{8+2\sqrt{15}}{4}+\frac{8-2\sqrt{15}}{64-2^{2}\left(\sqrt{15}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{8+2\sqrt{15}}{4}+\frac{8-2\sqrt{15}}{64-4\left(\sqrt{15}\right)^{2}}$$

The square of \sqrt{15} is 15.

$$\frac{8+2\sqrt{15}}{4}+\frac{8-2\sqrt{15}}{64-4\times 15}$$

Multiply 4 and 15 to get 60.

$$\frac{8+2\sqrt{15}}{4}+\frac{8-2\sqrt{15}}{64-60}$$

Subtract 60 from 64 to get 4.

$$\frac{8+2\sqrt{15}}{4}+\frac{8-2\sqrt{15}}{4}$$

Since \frac{8+2\sqrt{15}}{4} and \frac{8-2\sqrt{15}}{4} have the same denominator, add them by adding their numerators.

$$\frac{8+2\sqrt{15}+8-2\sqrt{15}}{4}$$

Do the calculations in 8+2\sqrt{15}+8-2\sqrt{15}.

$$\frac{16}{4}$$

Divide 16 by 4 to get 4.

$$4$$

Factor

$$2^{2}$$