Solve frac{10-sqrt{32}}{sqrt{2}}

$$\frac { 10 - \sqrt { 32 } } { \sqrt { 2 } }$$

Evaluate

$$5\sqrt{2}-4\approx 3.071067812$$

Solution Steps

Factor 32=4^{2}\times 2. Rewrite the square root of the product \sqrt{4^{2}\times 2} as the product of square roots \sqrt{4^{2}}\sqrt{2}. Take the square root of 4^{2}.

$$\frac{10-4\sqrt{2}}{\sqrt{2}}$$

Rationalize the denominator of \frac{10-4\sqrt{2}}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\left(10-4\sqrt{2}\right)\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(10-4\sqrt{2}\right)\sqrt{2}}{2}$$

Use the distributive property to multiply 10-4\sqrt{2} by \sqrt{2}.

$$\frac{10\sqrt{2}-4\left(\sqrt{2}\right)^{2}}{2}$$

The square of \sqrt{2} is 2.

$$\frac{10\sqrt{2}-4\times 2}{2}$$

Multiply -4 and 2 to get -8.

$$\frac{10\sqrt{2}-8}{2}$$

Divide each term of 10\sqrt{2}-8 by 2 to get 5\sqrt{2}-4.

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