Solve 11+31:(5+sqrt{5})^2

$$11 + 31 : ( 5 + \sqrt { 5 } ) ^ { 2 }$$

Evaluate

$$\frac{533-31\sqrt{5}}{40}\approx 11.592047317$$

Short Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(5+\sqrt{5}\right)^{2}.

$$11+\frac{31}{25+10\sqrt{5}+\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$11+\frac{31}{25+10\sqrt{5}+5}$$

Add 25 and 5 to get 30.

$$11+\frac{31}{30+10\sqrt{5}}$$

Rationalize the denominator of \frac{31}{30+10\sqrt{5}} by multiplying numerator and denominator by 30-10\sqrt{5}.

$$11+\frac{31\left(30-10\sqrt{5}\right)}{\left(30+10\sqrt{5}\right)\left(30-10\sqrt{5}\right)}$$

Consider \left(30+10\sqrt{5}\right)\left(30-10\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}.

$$11+\frac{31\left(30-10\sqrt{5}\right)}{30^{2}-\left(10\sqrt{5}\right)^{2}}$$

Calculate 30 to the power of 2 and get 900.

$$11+\frac{31\left(30-10\sqrt{5}\right)}{900-\left(10\sqrt{5}\right)^{2}}$$

Expand \left(10\sqrt{5}\right)^{2}.

$$11+\frac{31\left(30-10\sqrt{5}\right)}{900-10^{2}\left(\sqrt{5}\right)^{2}}$$

Calculate 10 to the power of 2 and get 100.

$$11+\frac{31\left(30-10\sqrt{5}\right)}{900-100\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$11+\frac{31\left(30-10\sqrt{5}\right)}{900-100\times 5}$$

Multiply 100 and 5 to get 500.

$$11+\frac{31\left(30-10\sqrt{5}\right)}{900-500}$$

Subtract 500 from 900 to get 400.

$$11+\frac{31\left(30-10\sqrt{5}\right)}{400}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 11 times \frac{400}{400}.

$$\frac{11\times 400}{400}+\frac{31\left(30-10\sqrt{5}\right)}{400}$$

Since \frac{11\times 400}{400} and \frac{31\left(30-10\sqrt{5}\right)}{400} have the same denominator, add them by adding their numerators.

$$\frac{11\times 400+31\left(30-10\sqrt{5}\right)}{400}$$

Do the multiplications in 11\times 400+31\left(30-10\sqrt{5}\right).

$$\frac{4400+930-310\sqrt{5}}{400}$$

Do the calculations in 4400+930-310\sqrt{5}.

$$\frac{5330-310\sqrt{5}}{400}$$

Expand

$$\frac{533 - 31 \sqrt{5}}{40} = 11.592047317437663$$

Short Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(5+\sqrt{5}\right)^{2}.

$$11+\frac{31}{25+10\sqrt{5}+\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$11+\frac{31}{25+10\sqrt{5}+5}$$

Add 25 and 5 to get 30.

$$11+\frac{31}{30+10\sqrt{5}}$$

Rationalize the denominator of \frac{31}{30+10\sqrt{5}} by multiplying numerator and denominator by 30-10\sqrt{5}.

$$11+\frac{31\left(30-10\sqrt{5}\right)}{\left(30+10\sqrt{5}\right)\left(30-10\sqrt{5}\right)}$$

Consider \left(30+10\sqrt{5}\right)\left(30-10\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}.

$$11+\frac{31\left(30-10\sqrt{5}\right)}{30^{2}-\left(10\sqrt{5}\right)^{2}}$$

Calculate 30 to the power of 2 and get 900.

$$11+\frac{31\left(30-10\sqrt{5}\right)}{900-\left(10\sqrt{5}\right)^{2}}$$

Expand \left(10\sqrt{5}\right)^{2}.

$$11+\frac{31\left(30-10\sqrt{5}\right)}{900-10^{2}\left(\sqrt{5}\right)^{2}}$$

Calculate 10 to the power of 2 and get 100.

$$11+\frac{31\left(30-10\sqrt{5}\right)}{900-100\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$11+\frac{31\left(30-10\sqrt{5}\right)}{900-100\times 5}$$

Multiply 100 and 5 to get 500.

$$11+\frac{31\left(30-10\sqrt{5}\right)}{900-500}$$

Subtract 500 from 900 to get 400.

$$11+\frac{31\left(30-10\sqrt{5}\right)}{400}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 11 times \frac{400}{400}.

$$\frac{11\times 400}{400}+\frac{31\left(30-10\sqrt{5}\right)}{400}$$

Since \frac{11\times 400}{400} and \frac{31\left(30-10\sqrt{5}\right)}{400} have the same denominator, add them by adding their numerators.

$$\frac{11\times 400+31\left(30-10\sqrt{5}\right)}{400}$$

Do the multiplications in 11\times 400+31\left(30-10\sqrt{5}\right).

$$\frac{4400+930-310\sqrt{5}}{400}$$

Do the calculations in 4400+930-310\sqrt{5}.

$$\frac{5330-310\sqrt{5}}{400}$$