Solve 3/sqrt{5+2}-frac{sqrt{2}}{2sqrt{2}-1}

$$\frac { 3 } { \sqrt { 5 } + 2 } - \frac { \sqrt { 2 } } { 2 \sqrt { 2 } - 1 }$$

Evaluate

$$-\frac{\sqrt{2}}{7}+3\sqrt{5}-\frac{46}{7}\approx -0.065255148$$

Short Solution Steps

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

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

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

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

Square \sqrt{5}. Square 2.

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

Subtract 4 from 5 to get 1.

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

Anything divided by one gives itself.

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

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

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

Consider \left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{2} is 2.

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

Multiply 4 and 2 to get 8.

$$3\left(\sqrt{5}-2\right)-\frac{\sqrt{2}\left(2\sqrt{2}+1\right)}{8-1^{2}}$$

Calculate 1 to the power of 2 and get 1.

$$3\left(\sqrt{5}-2\right)-\frac{\sqrt{2}\left(2\sqrt{2}+1\right)}{8-1}$$

Subtract 1 from 8 to get 7.

$$3\left(\sqrt{5}-2\right)-\frac{\sqrt{2}\left(2\sqrt{2}+1\right)}{7}$$

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

$$3\sqrt{5}-6-\frac{\sqrt{2}\left(2\sqrt{2}+1\right)}{7}$$

Use the distributive property to multiply \sqrt{2} by 2\sqrt{2}+1.

$$3\sqrt{5}-6-\frac{2\left(\sqrt{2}\right)^{2}+\sqrt{2}}{7}$$

The square of \sqrt{2} is 2.

$$3\sqrt{5}-6-\frac{2\times 2+\sqrt{2}}{7}$$

Multiply 2 and 2 to get 4.

$$3\sqrt{5}-6-\frac{4+\sqrt{2}}{7}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 3\sqrt{5}-6 times \frac{7}{7}.

$$\frac{7\left(3\sqrt{5}-6\right)}{7}-\frac{4+\sqrt{2}}{7}$$

Since \frac{7\left(3\sqrt{5}-6\right)}{7} and \frac{4+\sqrt{2}}{7} have the same denominator, subtract them by subtracting their numerators.

$$\frac{7\left(3\sqrt{5}-6\right)-\left(4+\sqrt{2}\right)}{7}$$

Do the multiplications in 7\left(3\sqrt{5}-6\right)-\left(4+\sqrt{2}\right).

$$\frac{21\sqrt{5}-42-4-\sqrt{2}}{7}$$

Do the calculations in 21\sqrt{5}-42-4-\sqrt{2}.

$$\frac{21\sqrt{5}-46-\sqrt{2}}{7}$$