Solve (1/2+sqrt{5}-1/2-sqrt{5})*frac{sqrt{121}}{sqrt{45}}:1/3

$$( \frac { 1 } { 2 + \sqrt { 5 } } - \frac { 1 } { 2 - \sqrt { 5 } } ) \cdot \frac { \sqrt { 121 } } { \sqrt { 45 } } : \frac { 1 } { 3 }$$

Evaluate

$$22$$

Solution Steps

Rationalize the denominator of \frac{1}{2+\sqrt{5}} by multiplying numerator and denominator by 2-\sqrt{5}.

$$\frac{\left(\frac{2-\sqrt{5}}{\left(2+\sqrt{5}\right)\left(2-\sqrt{5}\right)}-\frac{1}{2-\sqrt{5}}\right)\times \frac{\sqrt{121}}{\sqrt{45}}}{\frac{1}{3}}$$

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

$$\frac{\left(\frac{2-\sqrt{5}}{2^{2}-\left(\sqrt{5}\right)^{2}}-\frac{1}{2-\sqrt{5}}\right)\times \frac{\sqrt{121}}{\sqrt{45}}}{\frac{1}{3}}$$

Square 2. Square \sqrt{5}.

$$\frac{\left(\frac{2-\sqrt{5}}{4-5}-\frac{1}{2-\sqrt{5}}\right)\times \frac{\sqrt{121}}{\sqrt{45}}}{\frac{1}{3}}$$

Subtract 5 from 4 to get -1.

$$\frac{\left(\frac{2-\sqrt{5}}{-1}-\frac{1}{2-\sqrt{5}}\right)\times \frac{\sqrt{121}}{\sqrt{45}}}{\frac{1}{3}}$$

Anything divided by -1 gives its opposite. To find the opposite of 2-\sqrt{5}, find the opposite of each term.

$$\frac{\left(-2-\left(-\sqrt{5}\right)-\frac{1}{2-\sqrt{5}}\right)\times \frac{\sqrt{121}}{\sqrt{45}}}{\frac{1}{3}}$$

The opposite of -\sqrt{5} is \sqrt{5}.

$$\frac{\left(-2+\sqrt{5}-\frac{1}{2-\sqrt{5}}\right)\times \frac{\sqrt{121}}{\sqrt{45}}}{\frac{1}{3}}$$

Rationalize the denominator of \frac{1}{2-\sqrt{5}} by multiplying numerator and denominator by 2+\sqrt{5}.

$$\frac{\left(-2+\sqrt{5}-\frac{2+\sqrt{5}}{\left(2-\sqrt{5}\right)\left(2+\sqrt{5}\right)}\right)\times \frac{\sqrt{121}}{\sqrt{45}}}{\frac{1}{3}}$$

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

$$\frac{\left(-2+\sqrt{5}-\frac{2+\sqrt{5}}{2^{2}-\left(\sqrt{5}\right)^{2}}\right)\times \frac{\sqrt{121}}{\sqrt{45}}}{\frac{1}{3}}$$

Square 2. Square \sqrt{5}.

$$\frac{\left(-2+\sqrt{5}-\frac{2+\sqrt{5}}{4-5}\right)\times \frac{\sqrt{121}}{\sqrt{45}}}{\frac{1}{3}}$$

Subtract 5 from 4 to get -1.

$$\frac{\left(-2+\sqrt{5}-\frac{2+\sqrt{5}}{-1}\right)\times \frac{\sqrt{121}}{\sqrt{45}}}{\frac{1}{3}}$$

Anything divided by -1 gives its opposite. To find the opposite of 2+\sqrt{5}, find the opposite of each term.

$$\frac{\left(-2+\sqrt{5}-\left(-2-\sqrt{5}\right)\right)\times \frac{\sqrt{121}}{\sqrt{45}}}{\frac{1}{3}}$$

To find the opposite of -2-\sqrt{5}, find the opposite of each term.

$$\frac{\left(-2+\sqrt{5}-\left(-2\right)-\left(-\sqrt{5}\right)\right)\times \frac{\sqrt{121}}{\sqrt{45}}}{\frac{1}{3}}$$

The opposite of -2 is 2.

$$\frac{\left(-2+\sqrt{5}+2-\left(-\sqrt{5}\right)\right)\times \frac{\sqrt{121}}{\sqrt{45}}}{\frac{1}{3}}$$

The opposite of -\sqrt{5} is \sqrt{5}.

$$\frac{\left(-2+\sqrt{5}+2+\sqrt{5}\right)\times \frac{\sqrt{121}}{\sqrt{45}}}{\frac{1}{3}}$$

Add -2 and 2 to get 0.

$$\frac{\left(\sqrt{5}+\sqrt{5}\right)\times \frac{\sqrt{121}}{\sqrt{45}}}{\frac{1}{3}}$$

Combine \sqrt{5} and \sqrt{5} to get 2\sqrt{5}.

$$\frac{2\sqrt{5}\times \frac{\sqrt{121}}{\sqrt{45}}}{\frac{1}{3}}$$

Calculate the square root of 121 and get 11.

$$\frac{2\sqrt{5}\times \frac{11}{\sqrt{45}}}{\frac{1}{3}}$$

Factor 45=3^{2}\times 5. Rewrite the square root of the product \sqrt{3^{2}\times 5} as the product of square roots \sqrt{3^{2}}\sqrt{5}. Take the square root of 3^{2}.

$$\frac{2\sqrt{5}\times \frac{11}{3\sqrt{5}}}{\frac{1}{3}}$$

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

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

The square of \sqrt{5} is 5.

$$\frac{2\sqrt{5}\times \frac{11\sqrt{5}}{3\times 5}}{\frac{1}{3}}$$

Multiply 3 and 5 to get 15.

$$\frac{2\sqrt{5}\times \frac{11\sqrt{5}}{15}}{\frac{1}{3}}$$

Express 2\times \frac{11\sqrt{5}}{15} as a single fraction.

$$\frac{\frac{2\times 11\sqrt{5}}{15}\sqrt{5}}{\frac{1}{3}}$$

Multiply 2 and 11 to get 22.

$$\frac{\frac{22\sqrt{5}}{15}\sqrt{5}}{\frac{1}{3}}$$

Express \frac{22\sqrt{5}}{15}\sqrt{5} as a single fraction.

$$\frac{\frac{22\sqrt{5}\sqrt{5}}{15}}{\frac{1}{3}}$$

Divide \frac{22\sqrt{5}\sqrt{5}}{15} by \frac{1}{3} by multiplying \frac{22\sqrt{5}\sqrt{5}}{15} by the reciprocal of \frac{1}{3}.

$$\frac{22\sqrt{5}\sqrt{5}\times 3}{15}$$

Divide 22\sqrt{5}\sqrt{5}\times 3 by 15 to get 22\sqrt{5}\sqrt{5}\times \frac{1}{5}.

$$22\sqrt{5}\sqrt{5}\times \frac{1}{5}$$

Multiply \sqrt{5} and \sqrt{5} to get 5.

$$22\times 5\times \frac{1}{5}$$

Multiply 22 and 5 to get 110.

$$110\times \frac{1}{5}$$

Multiply 110 and \frac{1}{5} to get \frac{110}{5}.

$$\frac{110}{5}$$

Divide 110 by 5 to get 22.

$$22$$

Factor

$$2\times 11$$