Solve frac{-sqrt{14}}{1}+frac{sqrt{10}}{1}+frac{2sqrt{2}}{sqrt{5}+sqrt{7}}

$$\frac { - \sqrt { 14 } } { 1 } + \frac { \sqrt { 10 } } { 1 } + \frac { 2 \sqrt { 2 } } { \sqrt { 5 } + \sqrt { 7 } }$$

Evaluate

$$0$$

Solution Steps

Anything divided by one gives itself.

$$-\sqrt{14}+\frac{\sqrt{10}}{1}+\frac{2\sqrt{2}}{\sqrt{5}+\sqrt{7}}$$

Anything divided by one gives itself.

$$-\sqrt{14}+\sqrt{10}+\frac{2\sqrt{2}}{\sqrt{5}+\sqrt{7}}$$

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

$$-\sqrt{14}+\sqrt{10}+\frac{2\sqrt{2}\left(\sqrt{5}-\sqrt{7}\right)}{\left(\sqrt{5}+\sqrt{7}\right)\left(\sqrt{5}-\sqrt{7}\right)}$$

Consider \left(\sqrt{5}+\sqrt{7}\right)\left(\sqrt{5}-\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}.

$$-\sqrt{14}+\sqrt{10}+\frac{2\sqrt{2}\left(\sqrt{5}-\sqrt{7}\right)}{\left(\sqrt{5}\right)^{2}-\left(\sqrt{7}\right)^{2}}$$

Square \sqrt{5}. Square \sqrt{7}.

$$-\sqrt{14}+\sqrt{10}+\frac{2\sqrt{2}\left(\sqrt{5}-\sqrt{7}\right)}{5-7}$$

Subtract 7 from 5 to get -2.

$$-\sqrt{14}+\sqrt{10}+\frac{2\sqrt{2}\left(\sqrt{5}-\sqrt{7}\right)}{-2}$$

Cancel out -2 and -2.

$$-\sqrt{14}+\sqrt{10}-\sqrt{2}\left(\sqrt{5}-\sqrt{7}\right)$$

Use the distributive property to multiply -\sqrt{2} by \sqrt{5}-\sqrt{7}.

$$-\sqrt{14}+\sqrt{10}-\sqrt{2}\sqrt{5}+\sqrt{2}\sqrt{7}$$

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

$$-\sqrt{14}+\sqrt{10}-\sqrt{10}+\sqrt{2}\sqrt{7}$$

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

$$-\sqrt{14}+\sqrt{10}-\sqrt{10}+\sqrt{14}$$

Combine \sqrt{10} and -\sqrt{10} to get 0.

$$-\sqrt{14}+\sqrt{14}$$

Combine -\sqrt{14} and \sqrt{14} to get 0.

$$0$$

Factor

$$0$$