Solve (1-sqrt{18})(sqrt{2}+1/sqrt{2})

$$( 1 - \sqrt { 18 } ) ( \sqrt { 2 } + \frac { 1 } { \sqrt { 2 } } )$$

Evaluate

$$\frac{3\left(\sqrt{2}-6\right)}{2}\approx -6.878679656$$

Solution Steps

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}.

$$\left(1-3\sqrt{2}\right)\left(\sqrt{2}+\frac{1}{\sqrt{2}}\right)$$

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

$$\left(1-3\sqrt{2}\right)\left(\sqrt{2}+\frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}\right)$$

The square of \sqrt{2} is 2.

$$\left(1-3\sqrt{2}\right)\left(\sqrt{2}+\frac{\sqrt{2}}{2}\right)$$

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

$$\left(1-3\sqrt{2}\right)\times \frac{3}{2}\sqrt{2}$$

Use the distributive property to multiply 1-3\sqrt{2} by \frac{3}{2}.

$$\left(\frac{3}{2}-3\sqrt{2}\times \frac{3}{2}\right)\sqrt{2}$$

Express -3\times \frac{3}{2} as a single fraction.

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

Multiply -3 and 3 to get -9.

$$\left(\frac{3}{2}+\frac{-9}{2}\sqrt{2}\right)\sqrt{2}$$

Fraction \frac{-9}{2} can be rewritten as -\frac{9}{2} by extracting the negative sign.

$$\left(\frac{3}{2}-\frac{9}{2}\sqrt{2}\right)\sqrt{2}$$

Use the distributive property to multiply \frac{3}{2}-\frac{9}{2}\sqrt{2} by \sqrt{2}.

$$\frac{3}{2}\sqrt{2}-\frac{9}{2}\sqrt{2}\sqrt{2}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\frac{3}{2}\sqrt{2}-\frac{9}{2}\times 2$$

Cancel out 2 and 2.

$$\frac{3}{2}\sqrt{2}-9$$

Factor

$$\frac{3 {(\sqrt{2} - 6)}}{2} \approx -6.878679656$$