Solve frac{2-sqrt{5}}{2+3sqrt{5}}frac{2-3sqrt{5}}{2^2-{left(3sqrt{5}right)}^2}

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

Evaluate

$$\frac{158-73\sqrt{5}}{1681}\approx -0.003113006$$

Solution Steps

Rationalize the denominator of \frac{2-\sqrt{5}}{2+3\sqrt{5}} by multiplying numerator and denominator by 2-3\sqrt{5}.

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

Consider \left(2+3\sqrt{5}\right)\left(2-3\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}.

$$\frac{\left(2-\sqrt{5}\right)\left(2-3\sqrt{5}\right)}{2^{2}-\left(3\sqrt{5}\right)^{2}}\times \frac{2-3\sqrt{5}}{2^{2}-\left(3\sqrt{5}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\left(2-\sqrt{5}\right)\left(2-3\sqrt{5}\right)}{4-\left(3\sqrt{5}\right)^{2}}\times \frac{2-3\sqrt{5}}{2^{2}-\left(3\sqrt{5}\right)^{2}}$$

Expand \left(3\sqrt{5}\right)^{2}.

$$\frac{\left(2-\sqrt{5}\right)\left(2-3\sqrt{5}\right)}{4-3^{2}\left(\sqrt{5}\right)^{2}}\times \frac{2-3\sqrt{5}}{2^{2}-\left(3\sqrt{5}\right)^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{\left(2-\sqrt{5}\right)\left(2-3\sqrt{5}\right)}{4-9\left(\sqrt{5}\right)^{2}}\times \frac{2-3\sqrt{5}}{2^{2}-\left(3\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\left(2-\sqrt{5}\right)\left(2-3\sqrt{5}\right)}{4-9\times 5}\times \frac{2-3\sqrt{5}}{2^{2}-\left(3\sqrt{5}\right)^{2}}$$

Multiply 9 and 5 to get 45.

$$\frac{\left(2-\sqrt{5}\right)\left(2-3\sqrt{5}\right)}{4-45}\times \frac{2-3\sqrt{5}}{2^{2}-\left(3\sqrt{5}\right)^{2}}$$

Subtract 45 from 4 to get -41.

$$\frac{\left(2-\sqrt{5}\right)\left(2-3\sqrt{5}\right)}{-41}\times \frac{2-3\sqrt{5}}{2^{2}-\left(3\sqrt{5}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\left(2-\sqrt{5}\right)\left(2-3\sqrt{5}\right)}{-41}\times \frac{2-3\sqrt{5}}{4-\left(3\sqrt{5}\right)^{2}}$$

Expand \left(3\sqrt{5}\right)^{2}.

$$\frac{\left(2-\sqrt{5}\right)\left(2-3\sqrt{5}\right)}{-41}\times \frac{2-3\sqrt{5}}{4-3^{2}\left(\sqrt{5}\right)^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{\left(2-\sqrt{5}\right)\left(2-3\sqrt{5}\right)}{-41}\times \frac{2-3\sqrt{5}}{4-9\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\left(2-\sqrt{5}\right)\left(2-3\sqrt{5}\right)}{-41}\times \frac{2-3\sqrt{5}}{4-9\times 5}$$

Multiply 9 and 5 to get 45.

$$\frac{\left(2-\sqrt{5}\right)\left(2-3\sqrt{5}\right)}{-41}\times \frac{2-3\sqrt{5}}{4-45}$$

Subtract 45 from 4 to get -41.

$$\frac{\left(2-\sqrt{5}\right)\left(2-3\sqrt{5}\right)}{-41}\times \frac{2-3\sqrt{5}}{-41}$$

Multiply both numerator and denominator by -1.

$$\frac{\left(2-\sqrt{5}\right)\left(2-3\sqrt{5}\right)}{-41}\times \frac{-2+3\sqrt{5}}{41}$$

Multiply \frac{\left(2-\sqrt{5}\right)\left(2-3\sqrt{5}\right)}{-41} times \frac{-2+3\sqrt{5}}{41} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(2-\sqrt{5}\right)\left(2-3\sqrt{5}\right)\left(-2+3\sqrt{5}\right)}{-41\times 41}$$

Multiply -41 and 41 to get -1681.

$$\frac{\left(2-\sqrt{5}\right)\left(2-3\sqrt{5}\right)\left(-2+3\sqrt{5}\right)}{-1681}$$

Use the distributive property to multiply 2-\sqrt{5} by 2-3\sqrt{5} and combine like terms.

$$\frac{\left(4-8\sqrt{5}+3\left(\sqrt{5}\right)^{2}\right)\left(-2+3\sqrt{5}\right)}{-1681}$$

The square of \sqrt{5} is 5.

$$\frac{\left(4-8\sqrt{5}+3\times 5\right)\left(-2+3\sqrt{5}\right)}{-1681}$$

Multiply 3 and 5 to get 15.

$$\frac{\left(4-8\sqrt{5}+15\right)\left(-2+3\sqrt{5}\right)}{-1681}$$

Add 4 and 15 to get 19.

$$\frac{\left(19-8\sqrt{5}\right)\left(-2+3\sqrt{5}\right)}{-1681}$$

Use the distributive property to multiply 19-8\sqrt{5} by -2+3\sqrt{5} and combine like terms.

$$\frac{-38+73\sqrt{5}-24\left(\sqrt{5}\right)^{2}}{-1681}$$

The square of \sqrt{5} is 5.

$$\frac{-38+73\sqrt{5}-24\times 5}{-1681}$$

Multiply -24 and 5 to get -120.

$$\frac{-38+73\sqrt{5}-120}{-1681}$$

Subtract 120 from -38 to get -158.

$$\frac{-158+73\sqrt{5}}{-1681}$$

Multiply both numerator and denominator by -1.

$$\frac{158-73\sqrt{5}}{1681}$$