Solve 1/2sqrt{31+11}

$$\frac{ 1 }{ 2 \sqrt{ 31 } +11 }$$

Evaluate

$$\frac{2\sqrt{31}-11}{3}\approx 0.045176242$$

Solution Steps

Rationalize the denominator of \frac{1}{2\sqrt{31}+11} by multiplying numerator and denominator by 2\sqrt{31}-11.

$$\frac{2\sqrt{31}-11}{\left(2\sqrt{31}+11\right)\left(2\sqrt{31}-11\right)}$$

Consider \left(2\sqrt{31}+11\right)\left(2\sqrt{31}-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}.

$$\frac{2\sqrt{31}-11}{\left(2\sqrt{31}\right)^{2}-11^{2}}$$

Expand \left(2\sqrt{31}\right)^{2}.

$$\frac{2\sqrt{31}-11}{2^{2}\left(\sqrt{31}\right)^{2}-11^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{2\sqrt{31}-11}{4\left(\sqrt{31}\right)^{2}-11^{2}}$$

The square of \sqrt{31} is 31.

$$\frac{2\sqrt{31}-11}{4\times 31-11^{2}}$$

Multiply 4 and 31 to get 124.

$$\frac{2\sqrt{31}-11}{124-11^{2}}$$

Calculate 11 to the power of 2 and get 121.

$$\frac{2\sqrt{31}-11}{124-121}$$

Subtract 121 from 124 to get 3.

$$\frac{2\sqrt{31}-11}{3}$$