Solve frac{7sqrt{3}}{sqrt{10}+sqrt{3}}

$$\frac { 7 \sqrt { 3 } } { \sqrt { 10 } + \sqrt { 3 } }$$

Evaluate

$$\sqrt{30}-3\approx 2.477225575$$

Solution Steps

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

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

Consider \left(\sqrt{10}+\sqrt{3}\right)\left(\sqrt{10}-\sqrt{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}.

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

Square \sqrt{10}. Square \sqrt{3}.

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

Subtract 3 from 10 to get 7.

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

Use the distributive property to multiply 7\sqrt{3} by \sqrt{10}-\sqrt{3}.

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

To multiply \sqrt{3} and \sqrt{10}, multiply the numbers under the square root.

$$\frac{7\sqrt{30}-7\left(\sqrt{3}\right)^{2}}{7}$$

The square of \sqrt{3} is 3.

$$\frac{7\sqrt{30}-7\times 3}{7}$$

Multiply -7 and 3 to get -21.

$$\frac{7\sqrt{30}-21}{7}$$

Divide each term of 7\sqrt{30}-21 by 7 to get \sqrt{30}-3.

$$\sqrt{30}-3$$