Solve 2sqrt{32}-5sqrt{1/2}+6sqrt{1/8}

$$2 \sqrt{ 32 } -5 \sqrt{ \frac{ 1 }{ 2 } } +6 \sqrt{ \frac{ 1 }{ 8 } }$$

Evaluate

$$7\sqrt{2}\approx 9.899494937$$

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}.

$$2\times 4\sqrt{2}-5\sqrt{\frac{1}{2}}+6\sqrt{\frac{1}{8}}$$

Multiply 2 and 4 to get 8.

$$8\sqrt{2}-5\sqrt{\frac{1}{2}}+6\sqrt{\frac{1}{8}}$$

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

$$8\sqrt{2}-5\times \frac{\sqrt{1}}{\sqrt{2}}+6\sqrt{\frac{1}{8}}$$

Calculate the square root of 1 and get 1.

$$8\sqrt{2}-5\times \frac{1}{\sqrt{2}}+6\sqrt{\frac{1}{8}}$$

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

$$8\sqrt{2}-5\times \frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}+6\sqrt{\frac{1}{8}}$$

The square of \sqrt{2} is 2.

$$8\sqrt{2}-5\times \frac{\sqrt{2}}{2}+6\sqrt{\frac{1}{8}}$$

Express -5\times \frac{\sqrt{2}}{2} as a single fraction.

$$8\sqrt{2}+\frac{-5\sqrt{2}}{2}+6\sqrt{\frac{1}{8}}$$

Rewrite the square root of the division \sqrt{\frac{1}{8}} as the division of square roots \frac{\sqrt{1}}{\sqrt{8}}.

$$8\sqrt{2}+\frac{-5\sqrt{2}}{2}+6\times \frac{\sqrt{1}}{\sqrt{8}}$$

Calculate the square root of 1 and get 1.

$$8\sqrt{2}+\frac{-5\sqrt{2}}{2}+6\times \frac{1}{\sqrt{8}}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$8\sqrt{2}+\frac{-5\sqrt{2}}{2}+6\times \frac{1}{2\sqrt{2}}$$

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

$$8\sqrt{2}+\frac{-5\sqrt{2}}{2}+6\times \frac{\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$8\sqrt{2}+\frac{-5\sqrt{2}}{2}+6\times \frac{\sqrt{2}}{2\times 2}$$

Multiply 2 and 2 to get 4.

$$8\sqrt{2}+\frac{-5\sqrt{2}}{2}+6\times \frac{\sqrt{2}}{4}$$

Express 6\times \frac{\sqrt{2}}{4} as a single fraction.

$$8\sqrt{2}+\frac{-5\sqrt{2}}{2}+\frac{6\sqrt{2}}{4}$$

Combine 8\sqrt{2} and \frac{6\sqrt{2}}{4} to get \frac{19}{2}\sqrt{2}.

$$\frac{19}{2}\sqrt{2}+\frac{-5\sqrt{2}}{2}$$

Combine \frac{19}{2}\sqrt{2} and \frac{-5\sqrt{2}}{2} to get 7\sqrt{2}.

$$7\sqrt{2}$$