Solve 6/5+sqrt{11}

$$\frac { 6 } { 5 + \sqrt { 11 } }$$

Evaluate

$$\frac{15-3\sqrt{11}}{7}\approx 0.721446518$$

Solution Steps

Rationalize the denominator of \frac{6}{5+\sqrt{11}} by multiplying numerator and denominator by 5-\sqrt{11}.

$$\frac{6\left(5-\sqrt{11}\right)}{\left(5+\sqrt{11}\right)\left(5-\sqrt{11}\right)}$$

Consider \left(5+\sqrt{11}\right)\left(5-\sqrt{11}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{6\left(5-\sqrt{11}\right)}{5^{2}-\left(\sqrt{11}\right)^{2}}$$

Square 5. Square \sqrt{11}.

$$\frac{6\left(5-\sqrt{11}\right)}{25-11}$$

Subtract 11 from 25 to get 14.

$$\frac{6\left(5-\sqrt{11}\right)}{14}$$

Divide 6\left(5-\sqrt{11}\right) by 14 to get \frac{3}{7}\left(5-\sqrt{11}\right).

$$\frac{3}{7}\left(5-\sqrt{11}\right)$$

Use the distributive property to multiply \frac{3}{7} by 5-\sqrt{11}.

$$\frac{3}{7}\times 5+\frac{3}{7}\left(-1\right)\sqrt{11}$$

Express \frac{3}{7}\times 5 as a single fraction.

$$\frac{3\times 5}{7}+\frac{3}{7}\left(-1\right)\sqrt{11}$$

Multiply 3 and 5 to get 15.

$$\frac{15}{7}+\frac{3}{7}\left(-1\right)\sqrt{11}$$

Multiply \frac{3}{7} and -1 to get -\frac{3}{7}.

$$\frac{15}{7}-\frac{3}{7}\sqrt{11}$$