Solve {left(1/9-4sqrt{5}+1/9+4sqrt{5}right)}^2

$${ \left( \frac{ 1 }{ 9-4 \sqrt{ 5 } } + \frac{ 1 }{ 9+4 \sqrt{ 5 } } \right) }^{ 2 }$$

Evaluate

$$324$$

Solution Steps

Rationalize the denominator of \frac{1}{9-4\sqrt{5}} by multiplying numerator and denominator by 9+4\sqrt{5}.

$$\left(\frac{9+4\sqrt{5}}{\left(9-4\sqrt{5}\right)\left(9+4\sqrt{5}\right)}+\frac{1}{9+4\sqrt{5}}\right)^{2}$$

Consider \left(9-4\sqrt{5}\right)\left(9+4\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}.

$$\left(\frac{9+4\sqrt{5}}{9^{2}-\left(-4\sqrt{5}\right)^{2}}+\frac{1}{9+4\sqrt{5}}\right)^{2}$$

Calculate 9 to the power of 2 and get 81.

$$\left(\frac{9+4\sqrt{5}}{81-\left(-4\sqrt{5}\right)^{2}}+\frac{1}{9+4\sqrt{5}}\right)^{2}$$

Expand \left(-4\sqrt{5}\right)^{2}.

$$\left(\frac{9+4\sqrt{5}}{81-\left(-4\right)^{2}\left(\sqrt{5}\right)^{2}}+\frac{1}{9+4\sqrt{5}}\right)^{2}$$

Calculate -4 to the power of 2 and get 16.

$$\left(\frac{9+4\sqrt{5}}{81-16\left(\sqrt{5}\right)^{2}}+\frac{1}{9+4\sqrt{5}}\right)^{2}$$

The square of \sqrt{5} is 5.

$$\left(\frac{9+4\sqrt{5}}{81-16\times 5}+\frac{1}{9+4\sqrt{5}}\right)^{2}$$

Multiply 16 and 5 to get 80.

$$\left(\frac{9+4\sqrt{5}}{81-80}+\frac{1}{9+4\sqrt{5}}\right)^{2}$$

Subtract 80 from 81 to get 1.

$$\left(\frac{9+4\sqrt{5}}{1}+\frac{1}{9+4\sqrt{5}}\right)^{2}$$

Anything divided by one gives itself.

$$\left(9+4\sqrt{5}+\frac{1}{9+4\sqrt{5}}\right)^{2}$$

Rationalize the denominator of \frac{1}{9+4\sqrt{5}} by multiplying numerator and denominator by 9-4\sqrt{5}.

$$\left(9+4\sqrt{5}+\frac{9-4\sqrt{5}}{\left(9+4\sqrt{5}\right)\left(9-4\sqrt{5}\right)}\right)^{2}$$

Consider \left(9+4\sqrt{5}\right)\left(9-4\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}.

$$\left(9+4\sqrt{5}+\frac{9-4\sqrt{5}}{9^{2}-\left(4\sqrt{5}\right)^{2}}\right)^{2}$$

Calculate 9 to the power of 2 and get 81.

$$\left(9+4\sqrt{5}+\frac{9-4\sqrt{5}}{81-\left(4\sqrt{5}\right)^{2}}\right)^{2}$$

Expand \left(4\sqrt{5}\right)^{2}.

$$\left(9+4\sqrt{5}+\frac{9-4\sqrt{5}}{81-4^{2}\left(\sqrt{5}\right)^{2}}\right)^{2}$$

Calculate 4 to the power of 2 and get 16.

$$\left(9+4\sqrt{5}+\frac{9-4\sqrt{5}}{81-16\left(\sqrt{5}\right)^{2}}\right)^{2}$$

The square of \sqrt{5} is 5.

$$\left(9+4\sqrt{5}+\frac{9-4\sqrt{5}}{81-16\times 5}\right)^{2}$$

Multiply 16 and 5 to get 80.

$$\left(9+4\sqrt{5}+\frac{9-4\sqrt{5}}{81-80}\right)^{2}$$

Subtract 80 from 81 to get 1.

$$\left(9+4\sqrt{5}+\frac{9-4\sqrt{5}}{1}\right)^{2}$$

Anything divided by one gives itself.

$$\left(9+4\sqrt{5}+9-4\sqrt{5}\right)^{2}$$

Add 9 and 9 to get 18.

$$\left(18+4\sqrt{5}-4\sqrt{5}\right)^{2}$$

Combine 4\sqrt{5} and -4\sqrt{5} to get 0.

$$18^{2}$$

Calculate 18 to the power of 2 and get 324.

$$324$$

Factor

$$2^{2}\times 3^{4}$$