Solve (frac{1-sqrt{2}}{sqrt{2}}+frac{sqrt{2}-sqrt{3}}{sqrt{6}}+frac{sqrt{3}-sqrt{4}}{sqrt{12}}+frac{sqrt{4}-sqrt{5}}{sqrt{20}})*5/sqrt{5-5}

$$( \frac { 1 - \sqrt { 2 } } { \sqrt { 2 } } + \frac { \sqrt { 2 } - \sqrt { 3 } } { \sqrt { 6 } } + \frac { \sqrt { 3 } - \sqrt { 4 } } { \sqrt { 12 } } + \frac { \sqrt { 4 } - \sqrt { 5 } } { \sqrt { 20 } } ) \cdot \frac { 5 } { \sqrt { 5 } - 5 }$$

Evaluate

$$1$$

Short Solution Steps

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

$$\left(\frac{\left(1-\sqrt{2}\right)\sqrt{2}}{\left(\sqrt{2}\right)^{2}}+\frac{\sqrt{2}-\sqrt{3}}{\sqrt{6}}+\frac{\sqrt{3}-\sqrt{4}}{\sqrt{12}}+\frac{\sqrt{4}-\sqrt{5}}{\sqrt{20}}\right)\times \frac{5}{\sqrt{5}-5}$$

The square of \sqrt{2} is 2.

$$\left(\frac{\left(1-\sqrt{2}\right)\sqrt{2}}{2}+\frac{\sqrt{2}-\sqrt{3}}{\sqrt{6}}+\frac{\sqrt{3}-\sqrt{4}}{\sqrt{12}}+\frac{\sqrt{4}-\sqrt{5}}{\sqrt{20}}\right)\times \frac{5}{\sqrt{5}-5}$$

Rationalize the denominator of \frac{\sqrt{2}-\sqrt{3}}{\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$\left(\frac{\left(1-\sqrt{2}\right)\sqrt{2}}{2}+\frac{\left(\sqrt{2}-\sqrt{3}\right)\sqrt{6}}{\left(\sqrt{6}\right)^{2}}+\frac{\sqrt{3}-\sqrt{4}}{\sqrt{12}}+\frac{\sqrt{4}-\sqrt{5}}{\sqrt{20}}\right)\times \frac{5}{\sqrt{5}-5}$$

The square of \sqrt{6} is 6.

$$\left(\frac{\left(1-\sqrt{2}\right)\sqrt{2}}{2}+\frac{\left(\sqrt{2}-\sqrt{3}\right)\sqrt{6}}{6}+\frac{\sqrt{3}-\sqrt{4}}{\sqrt{12}}+\frac{\sqrt{4}-\sqrt{5}}{\sqrt{20}}\right)\times \frac{5}{\sqrt{5}-5}$$

Calculate the square root of 4 and get 2.

$$\left(\frac{\left(1-\sqrt{2}\right)\sqrt{2}}{2}+\frac{\left(\sqrt{2}-\sqrt{3}\right)\sqrt{6}}{6}+\frac{\sqrt{3}-2}{\sqrt{12}}+\frac{\sqrt{4}-\sqrt{5}}{\sqrt{20}}\right)\times \frac{5}{\sqrt{5}-5}$$

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

$$\left(\frac{\left(1-\sqrt{2}\right)\sqrt{2}}{2}+\frac{\left(\sqrt{2}-\sqrt{3}\right)\sqrt{6}}{6}+\frac{\sqrt{3}-2}{2\sqrt{3}}+\frac{\sqrt{4}-\sqrt{5}}{\sqrt{20}}\right)\times \frac{5}{\sqrt{5}-5}$$

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

$$\left(\frac{\left(1-\sqrt{2}\right)\sqrt{2}}{2}+\frac{\left(\sqrt{2}-\sqrt{3}\right)\sqrt{6}}{6}+\frac{\left(\sqrt{3}-2\right)\sqrt{3}}{2\left(\sqrt{3}\right)^{2}}+\frac{\sqrt{4}-\sqrt{5}}{\sqrt{20}}\right)\times \frac{5}{\sqrt{5}-5}$$

The square of \sqrt{3} is 3.

$$\left(\frac{\left(1-\sqrt{2}\right)\sqrt{2}}{2}+\frac{\left(\sqrt{2}-\sqrt{3}\right)\sqrt{6}}{6}+\frac{\left(\sqrt{3}-2\right)\sqrt{3}}{2\times 3}+\frac{\sqrt{4}-\sqrt{5}}{\sqrt{20}}\right)\times \frac{5}{\sqrt{5}-5}$$

Multiply 2 and 3 to get 6.

$$\left(\frac{\left(1-\sqrt{2}\right)\sqrt{2}}{2}+\frac{\left(\sqrt{2}-\sqrt{3}\right)\sqrt{6}}{6}+\frac{\left(\sqrt{3}-2\right)\sqrt{3}}{6}+\frac{\sqrt{4}-\sqrt{5}}{\sqrt{20}}\right)\times \frac{5}{\sqrt{5}-5}$$

Calculate the square root of 4 and get 2.

$$\left(\frac{\left(1-\sqrt{2}\right)\sqrt{2}}{2}+\frac{\left(\sqrt{2}-\sqrt{3}\right)\sqrt{6}}{6}+\frac{\left(\sqrt{3}-2\right)\sqrt{3}}{6}+\frac{2-\sqrt{5}}{\sqrt{20}}\right)\times \frac{5}{\sqrt{5}-5}$$

Factor 20=2^{2}\times 5. Rewrite the square root of the product \sqrt{2^{2}\times 5} as the product of square roots \sqrt{2^{2}}\sqrt{5}. Take the square root of 2^{2}.

$$\left(\frac{\left(1-\sqrt{2}\right)\sqrt{2}}{2}+\frac{\left(\sqrt{2}-\sqrt{3}\right)\sqrt{6}}{6}+\frac{\left(\sqrt{3}-2\right)\sqrt{3}}{6}+\frac{2-\sqrt{5}}{2\sqrt{5}}\right)\times \frac{5}{\sqrt{5}-5}$$

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

$$\left(\frac{\left(1-\sqrt{2}\right)\sqrt{2}}{2}+\frac{\left(\sqrt{2}-\sqrt{3}\right)\sqrt{6}}{6}+\frac{\left(\sqrt{3}-2\right)\sqrt{3}}{6}+\frac{\left(2-\sqrt{5}\right)\sqrt{5}}{2\left(\sqrt{5}\right)^{2}}\right)\times \frac{5}{\sqrt{5}-5}$$

The square of \sqrt{5} is 5.

$$\left(\frac{\left(1-\sqrt{2}\right)\sqrt{2}}{2}+\frac{\left(\sqrt{2}-\sqrt{3}\right)\sqrt{6}}{6}+\frac{\left(\sqrt{3}-2\right)\sqrt{3}}{6}+\frac{\left(2-\sqrt{5}\right)\sqrt{5}}{2\times 5}\right)\times \frac{5}{\sqrt{5}-5}$$

Multiply 2 and 5 to get 10.

$$\left(\frac{\left(1-\sqrt{2}\right)\sqrt{2}}{2}+\frac{\left(\sqrt{2}-\sqrt{3}\right)\sqrt{6}}{6}+\frac{\left(\sqrt{3}-2\right)\sqrt{3}}{6}+\frac{\left(2-\sqrt{5}\right)\sqrt{5}}{10}\right)\times \frac{5}{\sqrt{5}-5}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 6 is 6. Multiply \frac{\left(1-\sqrt{2}\right)\sqrt{2}}{2} times \frac{3}{3}.

$$\left(\frac{3\left(1-\sqrt{2}\right)\sqrt{2}}{6}+\frac{\left(\sqrt{2}-\sqrt{3}\right)\sqrt{6}}{6}+\frac{\left(\sqrt{3}-2\right)\sqrt{3}}{6}+\frac{\left(2-\sqrt{5}\right)\sqrt{5}}{10}\right)\times \frac{5}{\sqrt{5}-5}$$

Since \frac{3\left(1-\sqrt{2}\right)\sqrt{2}}{6} and \frac{\left(\sqrt{2}-\sqrt{3}\right)\sqrt{6}}{6} have the same denominator, add them by adding their numerators.

$$\left(\frac{3\left(1-\sqrt{2}\right)\sqrt{2}+\left(\sqrt{2}-\sqrt{3}\right)\sqrt{6}}{6}+\frac{\left(\sqrt{3}-2\right)\sqrt{3}}{6}+\frac{\left(2-\sqrt{5}\right)\sqrt{5}}{10}\right)\times \frac{5}{\sqrt{5}-5}$$

Do the multiplications in 3\left(1-\sqrt{2}\right)\sqrt{2}+\left(\sqrt{2}-\sqrt{3}\right)\sqrt{6}.

$$\left(\frac{3\sqrt{2}-6+2\sqrt{3}-3\sqrt{2}}{6}+\frac{\left(\sqrt{3}-2\right)\sqrt{3}}{6}+\frac{\left(2-\sqrt{5}\right)\sqrt{5}}{10}\right)\times \frac{5}{\sqrt{5}-5}$$

Do the calculations in 3\sqrt{2}-6+2\sqrt{3}-3\sqrt{2}.

$$\left(\frac{-6+2\sqrt{3}}{6}+\frac{\left(\sqrt{3}-2\right)\sqrt{3}}{6}+\frac{\left(2-\sqrt{5}\right)\sqrt{5}}{10}\right)\times \frac{5}{\sqrt{5}-5}$$

Since \frac{-6+2\sqrt{3}}{6} and \frac{\left(\sqrt{3}-2\right)\sqrt{3}}{6} have the same denominator, add them by adding their numerators.

$$\left(\frac{-6+2\sqrt{3}+\left(\sqrt{3}-2\right)\sqrt{3}}{6}+\frac{\left(2-\sqrt{5}\right)\sqrt{5}}{10}\right)\times \frac{5}{\sqrt{5}-5}$$

Do the multiplications in -6+2\sqrt{3}+\left(\sqrt{3}-2\right)\sqrt{3}.

$$\left(\frac{-6+2\sqrt{3}+3-2\sqrt{3}}{6}+\frac{\left(2-\sqrt{5}\right)\sqrt{5}}{10}\right)\times \frac{5}{\sqrt{5}-5}$$

Do the calculations in -6+2\sqrt{3}+3-2\sqrt{3}.

$$\left(\frac{-3}{6}+\frac{\left(2-\sqrt{5}\right)\sqrt{5}}{10}\right)\times \frac{5}{\sqrt{5}-5}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 6 and 10 is 30. Multiply \frac{-3}{6} times \frac{5}{5}. Multiply \frac{\left(2-\sqrt{5}\right)\sqrt{5}}{10} times \frac{3}{3}.

$$\left(\frac{-3\times 5}{30}+\frac{3\left(2-\sqrt{5}\right)\sqrt{5}}{30}\right)\times \frac{5}{\sqrt{5}-5}$$

Since \frac{-3\times 5}{30} and \frac{3\left(2-\sqrt{5}\right)\sqrt{5}}{30} have the same denominator, add them by adding their numerators.

$$\frac{-3\times 5+3\left(2-\sqrt{5}\right)\sqrt{5}}{30}\times \frac{5}{\sqrt{5}-5}$$

Do the multiplications in -3\times 5+3\left(2-\sqrt{5}\right)\sqrt{5}.

$$\frac{-15+6\sqrt{5}-15}{30}\times \frac{5}{\sqrt{5}-5}$$

Do the calculations in -15+6\sqrt{5}-15.

$$\frac{-30+6\sqrt{5}}{30}\times \frac{5}{\sqrt{5}-5}$$

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

$$\frac{-30+6\sqrt{5}}{30}\times \frac{5\left(\sqrt{5}+5\right)}{\left(\sqrt{5}-5\right)\left(\sqrt{5}+5\right)}$$

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

$$\frac{-30+6\sqrt{5}}{30}\times \frac{5\left(\sqrt{5}+5\right)}{\left(\sqrt{5}\right)^{2}-5^{2}}$$

Square \sqrt{5}. Square 5.

$$\frac{-30+6\sqrt{5}}{30}\times \frac{5\left(\sqrt{5}+5\right)}{5-25}$$

Subtract 25 from 5 to get -20.

$$\frac{-30+6\sqrt{5}}{30}\times \frac{5\left(\sqrt{5}+5\right)}{-20}$$

Divide 5\left(\sqrt{5}+5\right) by -20 to get -\frac{1}{4}\left(\sqrt{5}+5\right).

$$\frac{-30+6\sqrt{5}}{30}\left(-\frac{1}{4}\right)\left(\sqrt{5}+5\right)$$

Multiply \frac{-30+6\sqrt{5}}{30} times -\frac{1}{4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-\left(-30+6\sqrt{5}\right)}{30\times 4}\left(\sqrt{5}+5\right)$$

Express \frac{-\left(-30+6\sqrt{5}\right)}{30\times 4}\left(\sqrt{5}+5\right) as a single fraction.

$$\frac{-\left(-30+6\sqrt{5}\right)\left(\sqrt{5}+5\right)}{30\times 4}$$

Multiply 30 and 4 to get 120.

$$\frac{-\left(-30+6\sqrt{5}\right)\left(\sqrt{5}+5\right)}{120}$$

Use the distributive property to multiply -1 by -30+6\sqrt{5}.

$$\frac{\left(30-6\sqrt{5}\right)\left(\sqrt{5}+5\right)}{120}$$

Apply the distributive property by multiplying each term of 30-6\sqrt{5} by each term of \sqrt{5}+5.

$$\frac{30\sqrt{5}+150-6\left(\sqrt{5}\right)^{2}-30\sqrt{5}}{120}$$

The square of \sqrt{5} is 5.

$$\frac{30\sqrt{5}+150-6\times 5-30\sqrt{5}}{120}$$

Multiply -6 and 5 to get -30.

$$\frac{30\sqrt{5}+150-30-30\sqrt{5}}{120}$$

Subtract 30 from 150 to get 120.

$$\frac{30\sqrt{5}+120-30\sqrt{5}}{120}$$

Combine 30\sqrt{5} and -30\sqrt{5} to get 0.

$$\frac{120}{120}$$

Divide 120 by 120 to get 1.

$$1$$

Factor

$$1$$