Solve (sqrt{18}-sqrt{1/2})*8

$$( \sqrt{ 18 } - \sqrt{ \frac{ 1 }{ 2 } } ) \times 8$$

Evaluate

$$20\sqrt{2}\approx 28.284271247$$

Short 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}.

$$8\left(3\sqrt{2}-\sqrt{\frac{1}{2}}\right)$$

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

$$8\left(3\sqrt{2}-\frac{\sqrt{1}}{\sqrt{2}}\right)$$

Calculate the square root of 1 and get 1.

$$8\left(3\sqrt{2}-\frac{1}{\sqrt{2}}\right)$$

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

$$8\left(3\sqrt{2}-\frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}\right)$$

The square of \sqrt{2} is 2.

$$8\left(3\sqrt{2}-\frac{\sqrt{2}}{2}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 3\sqrt{2} times \frac{2}{2}.

$$8\left(\frac{2\times 3\sqrt{2}}{2}-\frac{\sqrt{2}}{2}\right)$$

Since \frac{2\times 3\sqrt{2}}{2} and \frac{\sqrt{2}}{2} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 2\times 3\sqrt{2}-\sqrt{2}.

$$8\times \frac{6\sqrt{2}-\sqrt{2}}{2}$$

Do the calculations in 6\sqrt{2}-\sqrt{2}.

$$8\times \frac{5\sqrt{2}}{2}$$

Cancel out 2, the greatest common factor in 8 and 2.

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

Multiply 4 and 5 to get 20.

$$20\sqrt{2}$$