Solve [(3-1/2)-2]^-1div(sqrt{18}-sqrt{8})

$$[ ( 3 - \frac { 1 } { 2 } ) - 2 ] ^ { - 1 } \div ( \sqrt { 18 } - \sqrt { 8 } )$$

Evaluate

$$\sqrt{2}\approx 1.414213562$$

Solution Steps

Subtract \frac{1}{2} from 3 to get \frac{5}{2}.

$$\frac{\left(\frac{5}{2}-2\right)^{-1}}{\sqrt{18}-\sqrt{8}}$$

Subtract 2 from \frac{5}{2} to get \frac{1}{2}.

$$\frac{\left(\frac{1}{2}\right)^{-1}}{\sqrt{18}-\sqrt{8}}$$

Calculate \frac{1}{2} to the power of -1 and get 2.

$$\frac{2}{\sqrt{18}-\sqrt{8}}$$

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

$$\frac{2}{3\sqrt{2}-\sqrt{8}}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$\frac{2}{3\sqrt{2}-2\sqrt{2}}$$

Combine 3\sqrt{2} and -2\sqrt{2} to get \sqrt{2}.

$$\frac{2}{\sqrt{2}}$$

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

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

The square of \sqrt{2} is 2.

$$\frac{2\sqrt{2}}{2}$$

Cancel out 2 and 2.

$$\sqrt{2}$$