Solve (2)2/sqrt{2}+sqrt{18}-4sqrt{1/2}

$$( 2 ) \frac { 2 } { \sqrt { 2 } } + \sqrt { 18 } - 4 \sqrt { \frac { 1 } { 2 } }$$

Evaluate

$$3\sqrt{2}\approx 4.242640687$$

Solution Steps

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

$$2\times \frac{2\sqrt{2}}{\left(\sqrt{2}\right)^{2}}+\sqrt{18}-4\sqrt{\frac{1}{2}}$$

The square of \sqrt{2} is 2.

$$2\times \frac{2\sqrt{2}}{2}+\sqrt{18}-4\sqrt{\frac{1}{2}}$$

Cancel out 2 and 2.

$$2\sqrt{2}+\sqrt{18}-4\sqrt{\frac{1}{2}}$$

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}.

$$2\sqrt{2}+3\sqrt{2}-4\sqrt{\frac{1}{2}}$$

Combine 2\sqrt{2} and 3\sqrt{2} to get 5\sqrt{2}.

$$5\sqrt{2}-4\sqrt{\frac{1}{2}}$$

Rewrite the square root of the division \sqrt{\frac{1}{2}} as the division of square roots \frac{\sqrt{1}}{\sqrt{2}}.

$$5\sqrt{2}-4\times \frac{\sqrt{1}}{\sqrt{2}}$$

Calculate the square root of 1 and get 1.

$$5\sqrt{2}-4\times \frac{1}{\sqrt{2}}$$

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

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

The square of \sqrt{2} is 2.

$$5\sqrt{2}-4\times \frac{\sqrt{2}}{2}$$

Cancel out 2, the greatest common factor in 4 and 2.

$$5\sqrt{2}-2\sqrt{2}$$

Combine 5\sqrt{2} and -2\sqrt{2} to get 3\sqrt{2}.

$$3\sqrt{2}$$