Solve f(x)=1/sqrt{2+sqrt{5}}-4/sqrt{3+sqrt{2}}+3/sqrt{5+sqrt{3}}

$$f ( x ) = \frac { 1 } { \sqrt { 2 } + \sqrt { 5 } } - \frac { 4 } { \sqrt { 3 } + \sqrt { 2 } } + \frac { 3 } { \sqrt { 5 } + \sqrt { 3 } }$$

Evaluate

$$\frac{11\left(\sqrt{5}+2\sqrt{2}-3\sqrt{3}\right)}{6}\approx -0.241371754$$

Short Solution Steps

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

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

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

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

Square \sqrt{2}. Square \sqrt{5}.

$$\frac{\sqrt{2}-\sqrt{5}}{2-5}-\frac{4}{\sqrt{3}+\sqrt{2}}+\frac{3}{\sqrt{5}+\sqrt{3}}$$

Subtract 5 from 2 to get -3.

$$\frac{\sqrt{2}-\sqrt{5}}{-3}-\frac{4}{\sqrt{3}+\sqrt{2}}+\frac{3}{\sqrt{5}+\sqrt{3}}$$

Multiply both numerator and denominator by -1.

$$\frac{-\sqrt{2}+\sqrt{5}}{3}-\frac{4}{\sqrt{3}+\sqrt{2}}+\frac{3}{\sqrt{5}+\sqrt{3}}$$

Rationalize the denominator of \frac{4}{\sqrt{3}+\sqrt{2}} by multiplying numerator and denominator by \sqrt{3}-\sqrt{2}.

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

Consider \left(\sqrt{3}+\sqrt{2}\right)\left(\sqrt{3}-\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square \sqrt{3}. Square \sqrt{2}.

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

Subtract 2 from 3 to get 1.

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

Anything divided by one gives itself.

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

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

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

Consider \left(\sqrt{5}+\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square \sqrt{5}. Square \sqrt{3}.

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

Subtract 3 from 5 to get 2.

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

Use the distributive property to multiply 4 by \sqrt{3}-\sqrt{2}.

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

To find the opposite of 4\sqrt{3}-4\sqrt{2}, find the opposite of each term.

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

The opposite of -4\sqrt{2} is 4\sqrt{2}.

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

Use the distributive property to multiply 3 by \sqrt{5}-\sqrt{3}.

$$\frac{-\sqrt{2}+\sqrt{5}}{3}-4\sqrt{3}+4\sqrt{2}+\frac{3\sqrt{5}-3\sqrt{3}}{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -4\sqrt{3}+4\sqrt{2} times \frac{3}{3}.

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

Since \frac{-\sqrt{2}+\sqrt{5}}{3} and \frac{3\left(-4\sqrt{3}+4\sqrt{2}\right)}{3} have the same denominator, add them by adding their numerators.

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

Do the multiplications in -\sqrt{2}+\sqrt{5}+3\left(-4\sqrt{3}+4\sqrt{2}\right).

$$\frac{-\sqrt{2}+\sqrt{5}-12\sqrt{3}+12\sqrt{2}}{3}+\frac{3\sqrt{5}-3\sqrt{3}}{2}$$

Do the calculations in -\sqrt{2}+\sqrt{5}-12\sqrt{3}+12\sqrt{2}.

$$\frac{11\sqrt{2}+\sqrt{5}-12\sqrt{3}}{3}+\frac{3\sqrt{5}-3\sqrt{3}}{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 2 is 6. Multiply \frac{11\sqrt{2}+\sqrt{5}-12\sqrt{3}}{3} times \frac{2}{2}. Multiply \frac{3\sqrt{5}-3\sqrt{3}}{2} times \frac{3}{3}.

$$\frac{2\left(11\sqrt{2}+\sqrt{5}-12\sqrt{3}\right)}{6}+\frac{3\left(3\sqrt{5}-3\sqrt{3}\right)}{6}$$

Since \frac{2\left(11\sqrt{2}+\sqrt{5}-12\sqrt{3}\right)}{6} and \frac{3\left(3\sqrt{5}-3\sqrt{3}\right)}{6} have the same denominator, add them by adding their numerators.

$$\frac{2\left(11\sqrt{2}+\sqrt{5}-12\sqrt{3}\right)+3\left(3\sqrt{5}-3\sqrt{3}\right)}{6}$$

Do the multiplications in 2\left(11\sqrt{2}+\sqrt{5}-12\sqrt{3}\right)+3\left(3\sqrt{5}-3\sqrt{3}\right).

$$\frac{22\sqrt{2}+2\sqrt{5}-24\sqrt{3}+9\sqrt{5}-9\sqrt{3}}{6}$$

Do the calculations in 22\sqrt{2}+2\sqrt{5}-24\sqrt{3}+9\sqrt{5}-9\sqrt{3}.

$$\frac{22\sqrt{2}+11\sqrt{5}-33\sqrt{3}}{6}$$