Solve (frac{sqrt{2}}{sqrt{2}-18})^2

$$( \frac { \sqrt { 2 } } { \sqrt { 2 } - 18 } ) ^ { 2 }$$

Evaluate

$$\frac{18\sqrt{2}+163}{25921}\approx 0.007270393$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{2}}{\sqrt{2}-18} by multiplying numerator and denominator by \sqrt{2}+18.

$$\left(\frac{\sqrt{2}\left(\sqrt{2}+18\right)}{\left(\sqrt{2}-18\right)\left(\sqrt{2}+18\right)}\right)^{2}$$

Consider \left(\sqrt{2}-18\right)\left(\sqrt{2}+18\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\left(\frac{\sqrt{2}\left(\sqrt{2}+18\right)}{\left(\sqrt{2}\right)^{2}-18^{2}}\right)^{2}$$

Square \sqrt{2}. Square 18.

$$\left(\frac{\sqrt{2}\left(\sqrt{2}+18\right)}{2-324}\right)^{2}$$

Subtract 324 from 2 to get -322.

$$\left(\frac{\sqrt{2}\left(\sqrt{2}+18\right)}{-322}\right)^{2}$$

To raise \frac{\sqrt{2}\left(\sqrt{2}+18\right)}{-322} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(\sqrt{2}\left(\sqrt{2}+18\right)\right)^{2}}{\left(-322\right)^{2}}$$

Expand \left(\sqrt{2}\left(\sqrt{2}+18\right)\right)^{2}.

$$\frac{\left(\sqrt{2}\right)^{2}\left(\sqrt{2}+18\right)^{2}}{\left(-322\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{2\left(\sqrt{2}+18\right)^{2}}{\left(-322\right)^{2}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{2}+18\right)^{2}.

$$\frac{2\left(\left(\sqrt{2}\right)^{2}+36\sqrt{2}+324\right)}{\left(-322\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{2\left(2+36\sqrt{2}+324\right)}{\left(-322\right)^{2}}$$

Add 2 and 324 to get 326.

$$\frac{2\left(326+36\sqrt{2}\right)}{\left(-322\right)^{2}}$$

Calculate -322 to the power of 2 and get 103684.

$$\frac{2\left(326+36\sqrt{2}\right)}{103684}$$

Divide 2\left(326+36\sqrt{2}\right) by 103684 to get \frac{1}{51842}\left(326+36\sqrt{2}\right).

$$\frac{1}{51842}\left(326+36\sqrt{2}\right)$$

Use the distributive property to multiply \frac{1}{51842} by 326+36\sqrt{2}.

$$\frac{163}{25921}+\frac{18}{25921}\sqrt{2}$$

Expand

$$\frac{18 \sqrt{2} + 163}{25921} = 0.007270392505023561$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{2}}{\sqrt{2}-18} by multiplying numerator and denominator by \sqrt{2}+18.

$$\left(\frac{\sqrt{2}\left(\sqrt{2}+18\right)}{\left(\sqrt{2}-18\right)\left(\sqrt{2}+18\right)}\right)^{2}$$

Consider \left(\sqrt{2}-18\right)\left(\sqrt{2}+18\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\left(\frac{\sqrt{2}\left(\sqrt{2}+18\right)}{\left(\sqrt{2}\right)^{2}-18^{2}}\right)^{2}$$

Square \sqrt{2}. Square 18.

$$\left(\frac{\sqrt{2}\left(\sqrt{2}+18\right)}{2-324}\right)^{2}$$

Subtract 324 from 2 to get -322.

$$\left(\frac{\sqrt{2}\left(\sqrt{2}+18\right)}{-322}\right)^{2}$$

To raise \frac{\sqrt{2}\left(\sqrt{2}+18\right)}{-322} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(\sqrt{2}\left(\sqrt{2}+18\right)\right)^{2}}{\left(-322\right)^{2}}$$

Expand \left(\sqrt{2}\left(\sqrt{2}+18\right)\right)^{2}.

$$\frac{\left(\sqrt{2}\right)^{2}\left(\sqrt{2}+18\right)^{2}}{\left(-322\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{2\left(\sqrt{2}+18\right)^{2}}{\left(-322\right)^{2}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{2}+18\right)^{2}.

$$\frac{2\left(\left(\sqrt{2}\right)^{2}+36\sqrt{2}+324\right)}{\left(-322\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{2\left(2+36\sqrt{2}+324\right)}{\left(-322\right)^{2}}$$

Add 2 and 324 to get 326.

$$\frac{2\left(326+36\sqrt{2}\right)}{\left(-322\right)^{2}}$$

Calculate -322 to the power of 2 and get 103684.

$$\frac{2\left(326+36\sqrt{2}\right)}{103684}$$

Divide 2\left(326+36\sqrt{2}\right) by 103684 to get \frac{1}{51842}\left(326+36\sqrt{2}\right).

$$\frac{1}{51842}\left(326+36\sqrt{2}\right)$$

Use the distributive property to multiply \frac{1}{51842} by 326+36\sqrt{2}.

$$\frac{163}{25921}+\frac{18}{25921}\sqrt{2}$$