Solve frac{sqrt{25}-16}{sqrt{25}-sqrt{10}}

$$\frac { \sqrt { 25 } - 16 } { \sqrt { 25 } - \sqrt { 10 } }$$

Evaluate

$$-\frac{11\sqrt{10}}{15}-\frac{11}{3}\approx -5.985670284$$

Solution Steps

Calculate the square root of 25 and get 5.

$$\frac{5-16}{\sqrt{25}-\sqrt{10}}$$

Subtract 16 from 5 to get -11.

$$\frac{-11}{\sqrt{25}-\sqrt{10}}$$

Calculate the square root of 25 and get 5.

$$\frac{-11}{5-\sqrt{10}}$$

Rationalize the denominator of \frac{-11}{5-\sqrt{10}} by multiplying numerator and denominator by 5+\sqrt{10}.

$$\frac{-11\left(5+\sqrt{10}\right)}{\left(5-\sqrt{10}\right)\left(5+\sqrt{10}\right)}$$

Consider \left(5-\sqrt{10}\right)\left(5+\sqrt{10}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{-11\left(5+\sqrt{10}\right)}{5^{2}-\left(\sqrt{10}\right)^{2}}$$

Square 5. Square \sqrt{10}.

$$\frac{-11\left(5+\sqrt{10}\right)}{25-10}$$

Subtract 10 from 25 to get 15.

$$\frac{-11\left(5+\sqrt{10}\right)}{15}$$

Use the distributive property to multiply -11 by 5+\sqrt{10}.

$$\frac{-55-11\sqrt{10}}{15}$$