Solve frac{3sqrt{2}-2sqrt{5}}{2sqrt{5}+5sqrt{2^2}}

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

Evaluate

$$\frac{3\sqrt{2}}{8}-\frac{\sqrt{5}}{4}-\frac{3\sqrt{10}}{40}+\frac{1}{4}\approx -0.015857733$$

Solution Steps

Calculate 2 to the power of 2 and get 4.

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

Calculate the square root of 4 and get 2.

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

Multiply 5 and 2 to get 10.

$$\frac{3\sqrt{2}-2\sqrt{5}}{2\sqrt{5}+10}$$

Rationalize the denominator of \frac{3\sqrt{2}-2\sqrt{5}}{2\sqrt{5}+10} by multiplying numerator and denominator by 2\sqrt{5}-10.

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

Consider \left(2\sqrt{5}+10\right)\left(2\sqrt{5}-10\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(3\sqrt{2}-2\sqrt{5}\right)\left(2\sqrt{5}-10\right)}{\left(2\sqrt{5}\right)^{2}-10^{2}}$$

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

$$\frac{\left(3\sqrt{2}-2\sqrt{5}\right)\left(2\sqrt{5}-10\right)}{2^{2}\left(\sqrt{5}\right)^{2}-10^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\left(3\sqrt{2}-2\sqrt{5}\right)\left(2\sqrt{5}-10\right)}{4\left(\sqrt{5}\right)^{2}-10^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\left(3\sqrt{2}-2\sqrt{5}\right)\left(2\sqrt{5}-10\right)}{4\times 5-10^{2}}$$

Multiply 4 and 5 to get 20.

$$\frac{\left(3\sqrt{2}-2\sqrt{5}\right)\left(2\sqrt{5}-10\right)}{20-10^{2}}$$

Calculate 10 to the power of 2 and get 100.

$$\frac{\left(3\sqrt{2}-2\sqrt{5}\right)\left(2\sqrt{5}-10\right)}{20-100}$$

Subtract 100 from 20 to get -80.

$$\frac{\left(3\sqrt{2}-2\sqrt{5}\right)\left(2\sqrt{5}-10\right)}{-80}$$

Apply the distributive property by multiplying each term of 3\sqrt{2}-2\sqrt{5} by each term of 2\sqrt{5}-10.

$$\frac{6\sqrt{2}\sqrt{5}-30\sqrt{2}-4\left(\sqrt{5}\right)^{2}+20\sqrt{5}}{-80}$$

To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{6\sqrt{10}-30\sqrt{2}-4\left(\sqrt{5}\right)^{2}+20\sqrt{5}}{-80}$$

The square of \sqrt{5} is 5.

$$\frac{6\sqrt{10}-30\sqrt{2}-4\times 5+20\sqrt{5}}{-80}$$

Multiply -4 and 5 to get -20.

$$\frac{6\sqrt{10}-30\sqrt{2}-20+20\sqrt{5}}{-80}$$