Solve sqrt{32}-3sqrt{1/2}+sqrt{25/2}

$$\sqrt { 32 } - 3 \sqrt { \frac { 1 } { 2 } } + \sqrt { \frac { 25 } { 2 } }$$

Evaluate

$$5\sqrt{2}\approx 7.071067812$$

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}.

$$4\sqrt{2}-3\sqrt{\frac{1}{2}}+\sqrt{\frac{25}{2}}$$

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

$$4\sqrt{2}-3\times \frac{\sqrt{1}}{\sqrt{2}}+\sqrt{\frac{25}{2}}$$

Calculate the square root of 1 and get 1.

$$4\sqrt{2}-3\times \frac{1}{\sqrt{2}}+\sqrt{\frac{25}{2}}$$

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

$$4\sqrt{2}-3\times \frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}+\sqrt{\frac{25}{2}}$$

The square of \sqrt{2} is 2.

$$4\sqrt{2}-3\times \frac{\sqrt{2}}{2}+\sqrt{\frac{25}{2}}$$

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

$$4\sqrt{2}+\frac{-3\sqrt{2}}{2}+\sqrt{\frac{25}{2}}$$

Rewrite the square root of the division \sqrt{\frac{25}{2}} as the division of square roots \frac{\sqrt{25}}{\sqrt{2}}.

$$4\sqrt{2}+\frac{-3\sqrt{2}}{2}+\frac{\sqrt{25}}{\sqrt{2}}$$

Calculate the square root of 25 and get 5.

$$4\sqrt{2}+\frac{-3\sqrt{2}}{2}+\frac{5}{\sqrt{2}}$$

Rationalize the denominator of \frac{5}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$4\sqrt{2}+\frac{-3\sqrt{2}}{2}+\frac{5\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$4\sqrt{2}+\frac{-3\sqrt{2}}{2}+\frac{5\sqrt{2}}{2}$$

Combine 4\sqrt{2} and \frac{5\sqrt{2}}{2} to get \frac{13}{2}\sqrt{2}.

$$\frac{13}{2}\sqrt{2}+\frac{-3\sqrt{2}}{2}$$

Combine \frac{13}{2}\sqrt{2} and \frac{-3\sqrt{2}}{2} to get 5\sqrt{2}.

$$5\sqrt{2}$$