Solve (1-8/sqrt{10})^2

$$( 1 - \frac { 8 } { \sqrt { 10 } } ) ^ { 2 }$$

Evaluate

$$\frac{37-8\sqrt{10}}{5}\approx 2.340355744$$

Solution Steps

Rationalize the denominator of \frac{8}{\sqrt{10}} by multiplying numerator and denominator by \sqrt{10}.

$$\left(1-\frac{8\sqrt{10}}{\left(\sqrt{10}\right)^{2}}\right)^{2}$$

The square of \sqrt{10} is 10.

$$\left(1-\frac{8\sqrt{10}}{10}\right)^{2}$$

Divide 8\sqrt{10} by 10 to get \frac{4}{5}\sqrt{10}.

$$\left(1-\frac{4}{5}\sqrt{10}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-\frac{4}{5}\sqrt{10}\right)^{2}.

$$1-\frac{8}{5}\sqrt{10}+\frac{16}{25}\left(\sqrt{10}\right)^{2}$$

The square of \sqrt{10} is 10.

$$1-\frac{8}{5}\sqrt{10}+\frac{16}{25}\times 10$$

Multiply \frac{16}{25} and 10 to get \frac{32}{5}.

$$1-\frac{8}{5}\sqrt{10}+\frac{32}{5}$$

Add 1 and \frac{32}{5} to get \frac{37}{5}.

$$\frac{37}{5}-\frac{8}{5}\sqrt{10}$$

Expand

$$\frac{37 - 8 \sqrt{10}}{5} = 2.340355743730593$$

Solution Steps

Rationalize the denominator of \frac{8}{\sqrt{10}} by multiplying numerator and denominator by \sqrt{10}.

$$\left(1-\frac{8\sqrt{10}}{\left(\sqrt{10}\right)^{2}}\right)^{2}$$

The square of \sqrt{10} is 10.

$$\left(1-\frac{8\sqrt{10}}{10}\right)^{2}$$

Divide 8\sqrt{10} by 10 to get \frac{4}{5}\sqrt{10}.

$$\left(1-\frac{4}{5}\sqrt{10}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-\frac{4}{5}\sqrt{10}\right)^{2}.

$$1-\frac{8}{5}\sqrt{10}+\frac{16}{25}\left(\sqrt{10}\right)^{2}$$

The square of \sqrt{10} is 10.

$$1-\frac{8}{5}\sqrt{10}+\frac{16}{25}\times 10$$

Multiply \frac{16}{25} and 10 to get \frac{32}{5}.

$$1-\frac{8}{5}\sqrt{10}+\frac{32}{5}$$

Add 1 and \frac{32}{5} to get \frac{37}{5}.

$$\frac{37}{5}-\frac{8}{5}\sqrt{10}$$