Solve 31+8sqrt{15}+1/31+8sqrt{15}

$$31+8 \sqrt{ 15 } + \frac{ 1 }{ 31+8 \sqrt{ 15 } }$$

Evaluate

$$62$$

Solution Steps

Rationalize the denominator of \frac{1}{31+8\sqrt{15}} by multiplying numerator and denominator by 31-8\sqrt{15}.

$$31+8\sqrt{15}+\frac{31-8\sqrt{15}}{\left(31+8\sqrt{15}\right)\left(31-8\sqrt{15}\right)}$$

Consider \left(31+8\sqrt{15}\right)\left(31-8\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}.

$$31+8\sqrt{15}+\frac{31-8\sqrt{15}}{31^{2}-\left(8\sqrt{15}\right)^{2}}$$

Calculate 31 to the power of 2 and get 961.

$$31+8\sqrt{15}+\frac{31-8\sqrt{15}}{961-\left(8\sqrt{15}\right)^{2}}$$

Expand \left(8\sqrt{15}\right)^{2}.

$$31+8\sqrt{15}+\frac{31-8\sqrt{15}}{961-8^{2}\left(\sqrt{15}\right)^{2}}$$

Calculate 8 to the power of 2 and get 64.

$$31+8\sqrt{15}+\frac{31-8\sqrt{15}}{961-64\left(\sqrt{15}\right)^{2}}$$

The square of \sqrt{15} is 15.

$$31+8\sqrt{15}+\frac{31-8\sqrt{15}}{961-64\times 15}$$

Multiply 64 and 15 to get 960.

$$31+8\sqrt{15}+\frac{31-8\sqrt{15}}{961-960}$$

Subtract 960 from 961 to get 1.

$$31+8\sqrt{15}+\frac{31-8\sqrt{15}}{1}$$

Anything divided by one gives itself.

$$31+8\sqrt{15}+31-8\sqrt{15}$$

Add 31 and 31 to get 62.

$$62+8\sqrt{15}-8\sqrt{15}$$

Combine 8\sqrt{15} and -8\sqrt{15} to get 0.

$$62$$

Factor

$$2\times 31$$