Solve frac{sqrt{5}+sqrt{2}}{sqrt{2}+sqrt{7}}

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

Evaluate

$$\frac{\sqrt{14}+\sqrt{35}-\sqrt{10}-2}{5}\approx 0.899091902$$

Solution Steps

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

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

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

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

Square \sqrt{2}. Square \sqrt{7}.

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

Subtract 7 from 2 to get -5.

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

Apply the distributive property by multiplying each term of \sqrt{5}+\sqrt{2} by each term of \sqrt{2}-\sqrt{7}.

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

To multiply \sqrt{5} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{\sqrt{10}-\sqrt{5}\sqrt{7}+\left(\sqrt{2}\right)^{2}-\sqrt{2}\sqrt{7}}{-5}$$

To multiply \sqrt{5} and \sqrt{7}, multiply the numbers under the square root.

$$\frac{\sqrt{10}-\sqrt{35}+\left(\sqrt{2}\right)^{2}-\sqrt{2}\sqrt{7}}{-5}$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{10}-\sqrt{35}+2-\sqrt{2}\sqrt{7}}{-5}$$

To multiply \sqrt{2} and \sqrt{7}, multiply the numbers under the square root.

$$\frac{\sqrt{10}-\sqrt{35}+2-\sqrt{14}}{-5}$$

Multiply both numerator and denominator by -1.

$$\frac{-\sqrt{10}+\sqrt{35}-2+\sqrt{14}}{5}$$