Solve frac{sqrt{23}-sqrt{37}}{sqrt{23}+sqrt{37}}

$$\frac { \sqrt { 23 } - \sqrt { 37 } } { \sqrt { 23 } + \sqrt { 37 } }$$

Evaluate

$$\frac{\sqrt{851}-30}{7}\approx -0.118299387$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{23}-\sqrt{37}}{\sqrt{23}+\sqrt{37}} by multiplying numerator and denominator by \sqrt{23}-\sqrt{37}.

$$\frac{\left(\sqrt{23}-\sqrt{37}\right)\left(\sqrt{23}-\sqrt{37}\right)}{\left(\sqrt{23}+\sqrt{37}\right)\left(\sqrt{23}-\sqrt{37}\right)}$$

Consider \left(\sqrt{23}+\sqrt{37}\right)\left(\sqrt{23}-\sqrt{37}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(\sqrt{23}-\sqrt{37}\right)\left(\sqrt{23}-\sqrt{37}\right)}{\left(\sqrt{23}\right)^{2}-\left(\sqrt{37}\right)^{2}}$$

Square \sqrt{23}. Square \sqrt{37}.

$$\frac{\left(\sqrt{23}-\sqrt{37}\right)\left(\sqrt{23}-\sqrt{37}\right)}{23-37}$$

Subtract 37 from 23 to get -14.

$$\frac{\left(\sqrt{23}-\sqrt{37}\right)\left(\sqrt{23}-\sqrt{37}\right)}{-14}$$

Multiply \sqrt{23}-\sqrt{37} and \sqrt{23}-\sqrt{37} to get \left(\sqrt{23}-\sqrt{37}\right)^{2}.

$$\frac{\left(\sqrt{23}-\sqrt{37}\right)^{2}}{-14}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{23}-\sqrt{37}\right)^{2}.

$$\frac{\left(\sqrt{23}\right)^{2}-2\sqrt{23}\sqrt{37}+\left(\sqrt{37}\right)^{2}}{-14}$$

The square of \sqrt{23} is 23.

$$\frac{23-2\sqrt{23}\sqrt{37}+\left(\sqrt{37}\right)^{2}}{-14}$$

To multiply \sqrt{23} and \sqrt{37}, multiply the numbers under the square root.

$$\frac{23-2\sqrt{851}+\left(\sqrt{37}\right)^{2}}{-14}$$

The square of \sqrt{37} is 37.

$$\frac{23-2\sqrt{851}+37}{-14}$$

Add 23 and 37 to get 60.

$$\frac{60-2\sqrt{851}}{-14}$$