Solve frac{7-2sqrt{10}}{sqrt{10}-3}

$$\frac{ 7-2 \sqrt{ 10 } }{ \sqrt{ 10 } -3 }$$

Evaluate

$$\sqrt{10}+1\approx 4.16227766$$

Solution Steps

Rationalize the denominator of \frac{7-2\sqrt{10}}{\sqrt{10}-3} by multiplying numerator and denominator by \sqrt{10}+3.

$$\frac{\left(7-2\sqrt{10}\right)\left(\sqrt{10}+3\right)}{\left(\sqrt{10}-3\right)\left(\sqrt{10}+3\right)}$$

Consider \left(\sqrt{10}-3\right)\left(\sqrt{10}+3\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(7-2\sqrt{10}\right)\left(\sqrt{10}+3\right)}{\left(\sqrt{10}\right)^{2}-3^{2}}$$

Square \sqrt{10}. Square 3.

$$\frac{\left(7-2\sqrt{10}\right)\left(\sqrt{10}+3\right)}{10-9}$$

Subtract 9 from 10 to get 1.

$$\frac{\left(7-2\sqrt{10}\right)\left(\sqrt{10}+3\right)}{1}$$

Anything divided by one gives itself.

$$\left(7-2\sqrt{10}\right)\left(\sqrt{10}+3\right)$$

Apply the distributive property by multiplying each term of 7-2\sqrt{10} by each term of \sqrt{10}+3.

$$7\sqrt{10}+21-2\left(\sqrt{10}\right)^{2}-6\sqrt{10}$$

The square of \sqrt{10} is 10.

$$7\sqrt{10}+21-2\times 10-6\sqrt{10}$$

Multiply -2 and 10 to get -20.

$$7\sqrt{10}+21-20-6\sqrt{10}$$

Subtract 20 from 21 to get 1.

$$7\sqrt{10}+1-6\sqrt{10}$$

Combine 7\sqrt{10} and -6\sqrt{10} to get \sqrt{10}.

$$\sqrt{10}+1$$