Solve 1/5sqrt{2-2sqrt{5}}-frac{5sqrt{2}+2sqrt{5}}{30}

$$\frac { 1 } { 5 \sqrt { 2 } - 2 \sqrt { 5 } } - \frac { 5 \sqrt { 2 } + 2 \sqrt { 5 } } { 30 }$$

Evaluate

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Solution Steps

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

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

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

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

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

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

Calculate 5 to the power of 2 and get 25.

$$\frac{5\sqrt{2}+2\sqrt{5}}{25\left(\sqrt{2}\right)^{2}-\left(-2\sqrt{5}\right)^{2}}-\frac{5\sqrt{2}+2\sqrt{5}}{30}$$

The square of \sqrt{2} is 2.

$$\frac{5\sqrt{2}+2\sqrt{5}}{25\times 2-\left(-2\sqrt{5}\right)^{2}}-\frac{5\sqrt{2}+2\sqrt{5}}{30}$$

Multiply 25 and 2 to get 50.

$$\frac{5\sqrt{2}+2\sqrt{5}}{50-\left(-2\sqrt{5}\right)^{2}}-\frac{5\sqrt{2}+2\sqrt{5}}{30}$$

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

$$\frac{5\sqrt{2}+2\sqrt{5}}{50-\left(-2\right)^{2}\left(\sqrt{5}\right)^{2}}-\frac{5\sqrt{2}+2\sqrt{5}}{30}$$

Calculate -2 to the power of 2 and get 4.

$$\frac{5\sqrt{2}+2\sqrt{5}}{50-4\left(\sqrt{5}\right)^{2}}-\frac{5\sqrt{2}+2\sqrt{5}}{30}$$

The square of \sqrt{5} is 5.

$$\frac{5\sqrt{2}+2\sqrt{5}}{50-4\times 5}-\frac{5\sqrt{2}+2\sqrt{5}}{30}$$

Multiply 4 and 5 to get 20.

$$\frac{5\sqrt{2}+2\sqrt{5}}{50-20}-\frac{5\sqrt{2}+2\sqrt{5}}{30}$$

Subtract 20 from 50 to get 30.

$$\frac{5\sqrt{2}+2\sqrt{5}}{30}-\frac{5\sqrt{2}+2\sqrt{5}}{30}$$

Combine \frac{5\sqrt{2}+2\sqrt{5}}{30} and -\frac{5\sqrt{2}+2\sqrt{5}}{30} to get 0.

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Factor

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