Solve 1/2+sqrt{53}

$$\frac { 1 } { 2 + \sqrt { 53 } }$$

Evaluate

$$\frac{\sqrt{53}-2}{49}\approx 0.107757345$$

Solution Steps

Rationalize the denominator of \frac{1}{2+\sqrt{53}} by multiplying numerator and denominator by 2-\sqrt{53}.

$$\frac{2-\sqrt{53}}{\left(2+\sqrt{53}\right)\left(2-\sqrt{53}\right)}$$

Consider \left(2+\sqrt{53}\right)\left(2-\sqrt{53}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{2-\sqrt{53}}{2^{2}-\left(\sqrt{53}\right)^{2}}$$

Square 2. Square \sqrt{53}.

$$\frac{2-\sqrt{53}}{4-53}$$

Subtract 53 from 4 to get -49.

$$\frac{2-\sqrt{53}}{-49}$$

Multiply both numerator and denominator by -1.

$$\frac{-2+\sqrt{53}}{49}$$