Solve 9/5+sqrt{10}

$$\frac { 9 } { 5 + \sqrt { 10 } }$$

Evaluate

$$-\frac{3\sqrt{10}}{5}+3\approx 1.102633404$$

Solution Steps

Rationalize the denominator of \frac{9}{5+\sqrt{10}} by multiplying numerator and denominator by 5-\sqrt{10}.

$$\frac{9\left(5-\sqrt{10}\right)}{\left(5+\sqrt{10}\right)\left(5-\sqrt{10}\right)}$$

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}.

$$\frac{9\left(5-\sqrt{10}\right)}{5^{2}-\left(\sqrt{10}\right)^{2}}$$

Square 5. Square \sqrt{10}.

$$\frac{9\left(5-\sqrt{10}\right)}{25-10}$$

Subtract 10 from 25 to get 15.

$$\frac{9\left(5-\sqrt{10}\right)}{15}$$

Divide 9\left(5-\sqrt{10}\right) by 15 to get \frac{3}{5}\left(5-\sqrt{10}\right).

$$\frac{3}{5}\left(5-\sqrt{10}\right)$$

Use the distributive property to multiply \frac{3}{5} by 5-\sqrt{10}.

$$\frac{3}{5}\times 5+\frac{3}{5}\left(-1\right)\sqrt{10}$$

Cancel out 5 and 5.

$$3+\frac{3}{5}\left(-1\right)\sqrt{10}$$

Multiply \frac{3}{5} and -1 to get -\frac{3}{5}.

$$3-\frac{3}{5}\sqrt{10}$$