Solve 1-sqrt{3}+sqrt{5}-11/14sqrt{3+sqrt{5}}

$$1 - \sqrt { 3 } + \sqrt { 5 } - \frac { 11 } { 14 \sqrt { 3 } + \sqrt { 5 } }$$

Evaluate

$$\frac{54\sqrt{5}}{53}-\frac{67\sqrt{3}}{53}+1\approx 1.088684277$$

Solution Steps

Rationalize the denominator of \frac{11}{14\sqrt{3}+\sqrt{5}} by multiplying numerator and denominator by 14\sqrt{3}-\sqrt{5}.

$$1-\sqrt{3}+\sqrt{5}-\frac{11\left(14\sqrt{3}-\sqrt{5}\right)}{\left(14\sqrt{3}+\sqrt{5}\right)\left(14\sqrt{3}-\sqrt{5}\right)}$$

Consider \left(14\sqrt{3}+\sqrt{5}\right)\left(14\sqrt{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}.

$$1-\sqrt{3}+\sqrt{5}-\frac{11\left(14\sqrt{3}-\sqrt{5}\right)}{\left(14\sqrt{3}\right)^{2}-\left(\sqrt{5}\right)^{2}}$$

Expand \left(14\sqrt{3}\right)^{2}.

$$1-\sqrt{3}+\sqrt{5}-\frac{11\left(14\sqrt{3}-\sqrt{5}\right)}{14^{2}\left(\sqrt{3}\right)^{2}-\left(\sqrt{5}\right)^{2}}$$

Calculate 14 to the power of 2 and get 196.

$$1-\sqrt{3}+\sqrt{5}-\frac{11\left(14\sqrt{3}-\sqrt{5}\right)}{196\left(\sqrt{3}\right)^{2}-\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$1-\sqrt{3}+\sqrt{5}-\frac{11\left(14\sqrt{3}-\sqrt{5}\right)}{196\times 3-\left(\sqrt{5}\right)^{2}}$$

Multiply 196 and 3 to get 588.

$$1-\sqrt{3}+\sqrt{5}-\frac{11\left(14\sqrt{3}-\sqrt{5}\right)}{588-\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$1-\sqrt{3}+\sqrt{5}-\frac{11\left(14\sqrt{3}-\sqrt{5}\right)}{588-5}$$

Subtract 5 from 588 to get 583.

$$1-\sqrt{3}+\sqrt{5}-\frac{11\left(14\sqrt{3}-\sqrt{5}\right)}{583}$$

Divide 11\left(14\sqrt{3}-\sqrt{5}\right) by 583 to get \frac{1}{53}\left(14\sqrt{3}-\sqrt{5}\right).

$$1-\sqrt{3}+\sqrt{5}-\frac{1}{53}\left(14\sqrt{3}-\sqrt{5}\right)$$

Use the distributive property to multiply \frac{1}{53} by 14\sqrt{3}-\sqrt{5}.

$$1-\sqrt{3}+\sqrt{5}-\left(\frac{1}{53}\times 14\sqrt{3}+\frac{1}{53}\left(-1\right)\sqrt{5}\right)$$

Multiply \frac{1}{53} and 14 to get \frac{14}{53}.

$$1-\sqrt{3}+\sqrt{5}-\left(\frac{14}{53}\sqrt{3}+\frac{1}{53}\left(-1\right)\sqrt{5}\right)$$

Multiply \frac{1}{53} and -1 to get -\frac{1}{53}.

$$1-\sqrt{3}+\sqrt{5}-\left(\frac{14}{53}\sqrt{3}-\frac{1}{53}\sqrt{5}\right)$$

To find the opposite of \frac{14}{53}\sqrt{3}-\frac{1}{53}\sqrt{5}, find the opposite of each term.

$$1-\sqrt{3}+\sqrt{5}-\frac{14}{53}\sqrt{3}-\left(-\frac{1}{53}\sqrt{5}\right)$$

The opposite of -\frac{1}{53}\sqrt{5} is \frac{1}{53}\sqrt{5}.

$$1-\sqrt{3}+\sqrt{5}-\frac{14}{53}\sqrt{3}+\frac{1}{53}\sqrt{5}$$

Combine -\sqrt{3} and -\frac{14}{53}\sqrt{3} to get -\frac{67}{53}\sqrt{3}.

$$1-\frac{67}{53}\sqrt{3}+\sqrt{5}+\frac{1}{53}\sqrt{5}$$

Combine \sqrt{5} and \frac{1}{53}\sqrt{5} to get \frac{54}{53}\sqrt{5}.

$$1-\frac{67}{53}\sqrt{3}+\frac{54}{53}\sqrt{5}$$