Solve frac{sqrt{14}-sqrt{15}}{sqrt{14}+sqrt{15}}

$$\frac { \sqrt { 14 } - \sqrt { 15 } } { \sqrt { 14 } + \sqrt { 15 } }$$

Evaluate

$$2\sqrt{210}-29\approx -0.017246508$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{14}-\sqrt{15}}{\sqrt{14}+\sqrt{15}} by multiplying numerator and denominator by \sqrt{14}-\sqrt{15}.

$$\frac{\left(\sqrt{14}-\sqrt{15}\right)\left(\sqrt{14}-\sqrt{15}\right)}{\left(\sqrt{14}+\sqrt{15}\right)\left(\sqrt{14}-\sqrt{15}\right)}$$

Consider \left(\sqrt{14}+\sqrt{15}\right)\left(\sqrt{14}-\sqrt{15}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(\sqrt{14}-\sqrt{15}\right)\left(\sqrt{14}-\sqrt{15}\right)}{\left(\sqrt{14}\right)^{2}-\left(\sqrt{15}\right)^{2}}$$

Square \sqrt{14}. Square \sqrt{15}.

$$\frac{\left(\sqrt{14}-\sqrt{15}\right)\left(\sqrt{14}-\sqrt{15}\right)}{14-15}$$

Subtract 15 from 14 to get -1.

$$\frac{\left(\sqrt{14}-\sqrt{15}\right)\left(\sqrt{14}-\sqrt{15}\right)}{-1}$$

Multiply \sqrt{14}-\sqrt{15} and \sqrt{14}-\sqrt{15} to get \left(\sqrt{14}-\sqrt{15}\right)^{2}.

$$\frac{\left(\sqrt{14}-\sqrt{15}\right)^{2}}{-1}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{14}-\sqrt{15}\right)^{2}.

$$\frac{\left(\sqrt{14}\right)^{2}-2\sqrt{14}\sqrt{15}+\left(\sqrt{15}\right)^{2}}{-1}$$

The square of \sqrt{14} is 14.

$$\frac{14-2\sqrt{14}\sqrt{15}+\left(\sqrt{15}\right)^{2}}{-1}$$

To multiply \sqrt{14} and \sqrt{15}, multiply the numbers under the square root.

$$\frac{14-2\sqrt{210}+\left(\sqrt{15}\right)^{2}}{-1}$$

The square of \sqrt{15} is 15.

$$\frac{14-2\sqrt{210}+15}{-1}$$

Add 14 and 15 to get 29.

$$\frac{29-2\sqrt{210}}{-1}$$

Anything divided by -1 gives its opposite. To find the opposite of 29-2\sqrt{210}, find the opposite of each term.

$$-29-\left(-2\sqrt{210}\right)$$

The opposite of -2\sqrt{210} is 2\sqrt{210}.

$$-29+2\sqrt{210}$$