Solve frac{2sqrt{7}}{3-sqrt{5}}=

$$\frac { 2 \sqrt { 7 } } { 3 - \sqrt { 5 } } =$$

Evaluate

$$\frac{\sqrt{7}\left(\sqrt{5}+3\right)}{2}\approx 6.926666858$$

Solution Steps

Rationalize the denominator of \frac{2\sqrt{7}}{3-\sqrt{5}} by multiplying numerator and denominator by 3+\sqrt{5}.

$$\frac{2\sqrt{7}\left(3+\sqrt{5}\right)}{\left(3-\sqrt{5}\right)\left(3+\sqrt{5}\right)}$$

Consider \left(3-\sqrt{5}\right)\left(3+\sqrt{5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{2\sqrt{7}\left(3+\sqrt{5}\right)}{3^{2}-\left(\sqrt{5}\right)^{2}}$$

Square 3. Square \sqrt{5}.

$$\frac{2\sqrt{7}\left(3+\sqrt{5}\right)}{9-5}$$

Subtract 5 from 9 to get 4.

$$\frac{2\sqrt{7}\left(3+\sqrt{5}\right)}{4}$$

Use the distributive property to multiply 2\sqrt{7} by 3+\sqrt{5}.

$$\frac{6\sqrt{7}+2\sqrt{7}\sqrt{5}}{4}$$

To multiply \sqrt{7} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{6\sqrt{7}+2\sqrt{35}}{4}$$