Solve frac{{left(sqrt{5}+1-1div6right)}^2}{{left(sqrt{5}+6right)}^2}=

$$\frac{ { \left( \sqrt{ 5 } +1-1 \div 6 \right) }^{ 2 } }{ { \left( \sqrt{ 5 } +6 \right) }^{ 2 } } =$$

Evaluate

$$\frac{5}{36}\approx 0.138888889$$

Solution Steps

Subtract \frac{1}{6} from 1 to get \frac{5}{6}.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{5}+\frac{5}{6}\right)^{2}.

$$\frac{\left(\sqrt{5}\right)^{2}+\frac{5}{3}\sqrt{5}+\frac{25}{36}}{\left(\sqrt{5}+6\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{5+\frac{5}{3}\sqrt{5}+\frac{25}{36}}{\left(\sqrt{5}+6\right)^{2}}$$

Add 5 and \frac{25}{36} to get \frac{205}{36}.

$$\frac{\frac{205}{36}+\frac{5}{3}\sqrt{5}}{\left(\sqrt{5}+6\right)^{2}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{5}+6\right)^{2}.

$$\frac{\frac{205}{36}+\frac{5}{3}\sqrt{5}}{\left(\sqrt{5}\right)^{2}+12\sqrt{5}+36}$$

The square of \sqrt{5} is 5.

$$\frac{\frac{205}{36}+\frac{5}{3}\sqrt{5}}{5+12\sqrt{5}+36}$$

Add 5 and 36 to get 41.

$$\frac{\frac{205}{36}+\frac{5}{3}\sqrt{5}}{41+12\sqrt{5}}$$

Rationalize the denominator of \frac{\frac{205}{36}+\frac{5}{3}\sqrt{5}}{41+12\sqrt{5}} by multiplying numerator and denominator by 41-12\sqrt{5}.

$$\frac{\left(\frac{205}{36}+\frac{5}{3}\sqrt{5}\right)\left(41-12\sqrt{5}\right)}{\left(41+12\sqrt{5}\right)\left(41-12\sqrt{5}\right)}$$

Consider \left(41+12\sqrt{5}\right)\left(41-12\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}.

$$\frac{\left(\frac{205}{36}+\frac{5}{3}\sqrt{5}\right)\left(41-12\sqrt{5}\right)}{41^{2}-\left(12\sqrt{5}\right)^{2}}$$

Calculate 41 to the power of 2 and get 1681.

$$\frac{\left(\frac{205}{36}+\frac{5}{3}\sqrt{5}\right)\left(41-12\sqrt{5}\right)}{1681-\left(12\sqrt{5}\right)^{2}}$$

Expand \left(12\sqrt{5}\right)^{2}.

$$\frac{\left(\frac{205}{36}+\frac{5}{3}\sqrt{5}\right)\left(41-12\sqrt{5}\right)}{1681-12^{2}\left(\sqrt{5}\right)^{2}}$$

Calculate 12 to the power of 2 and get 144.

$$\frac{\left(\frac{205}{36}+\frac{5}{3}\sqrt{5}\right)\left(41-12\sqrt{5}\right)}{1681-144\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\left(\frac{205}{36}+\frac{5}{3}\sqrt{5}\right)\left(41-12\sqrt{5}\right)}{1681-144\times 5}$$

Multiply 144 and 5 to get 720.

$$\frac{\left(\frac{205}{36}+\frac{5}{3}\sqrt{5}\right)\left(41-12\sqrt{5}\right)}{1681-720}$$

Subtract 720 from 1681 to get 961.

$$\frac{\left(\frac{205}{36}+\frac{5}{3}\sqrt{5}\right)\left(41-12\sqrt{5}\right)}{961}$$

Use the distributive property to multiply \frac{205}{36}+\frac{5}{3}\sqrt{5} by 41-12\sqrt{5} and combine like terms.

$$\frac{\frac{8405}{36}-20\left(\sqrt{5}\right)^{2}}{961}$$

The square of \sqrt{5} is 5.

$$\frac{\frac{8405}{36}-20\times 5}{961}$$

Multiply -20 and 5 to get -100.

$$\frac{\frac{8405}{36}-100}{961}$$

Subtract 100 from \frac{8405}{36} to get \frac{4805}{36}.

$$\frac{\frac{4805}{36}}{961}$$

Express \frac{\frac{4805}{36}}{961} as a single fraction.

$$\frac{4805}{36\times 961}$$

Multiply 36 and 961 to get 34596.

$$\frac{4805}{34596}$$

Reduce the fraction \frac{4805}{34596} to lowest terms by extracting and canceling out 961.

$$\frac{5}{36}$$

Factor

$$\frac{5}{2 ^ {2} \cdot 3 ^ {2}} = 0.1388888888888889$$