Solve 2/sqrt{7+5}

$$\frac{ 2 }{ \sqrt{ 7 } +5 }$$

Evaluate

$$\frac{5-\sqrt{7}}{9}\approx 0.261583188$$

Solution Steps

Rationalize the denominator of \frac{2}{\sqrt{7}+5} by multiplying numerator and denominator by \sqrt{7}-5.

$$\frac{2\left(\sqrt{7}-5\right)}{\left(\sqrt{7}+5\right)\left(\sqrt{7}-5\right)}$$

Consider \left(\sqrt{7}+5\right)\left(\sqrt{7}-5\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{2\left(\sqrt{7}-5\right)}{\left(\sqrt{7}\right)^{2}-5^{2}}$$

Square \sqrt{7}. Square 5.

$$\frac{2\left(\sqrt{7}-5\right)}{7-25}$$

Subtract 25 from 7 to get -18.

$$\frac{2\left(\sqrt{7}-5\right)}{-18}$$

Divide 2\left(\sqrt{7}-5\right) by -18 to get -\frac{1}{9}\left(\sqrt{7}-5\right).

$$-\frac{1}{9}\left(\sqrt{7}-5\right)$$

Use the distributive property to multiply -\frac{1}{9} by \sqrt{7}-5.

$$-\frac{1}{9}\sqrt{7}-\frac{1}{9}\left(-5\right)$$

Express -\frac{1}{9}\left(-5\right) as a single fraction.

$$-\frac{1}{9}\sqrt{7}+\frac{-\left(-5\right)}{9}$$

Multiply -1 and -5 to get 5.

$$-\frac{1}{9}\sqrt{7}+\frac{5}{9}$$