Solve 3sqrt{8}-1/2sqrt{32}+7sqrt{1/8}

$$3 \sqrt{ 8 } - \frac{ 1 }{ 2 } \sqrt{ 32 } +7 \sqrt{ \frac{ 1 }{ 8 } }$$

Evaluate

$$\frac{23\sqrt{2}}{4}\approx 8.131727984$$

Solution Steps

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}.

$$3\times 2\sqrt{2}-\frac{1}{2}\sqrt{32}+7\sqrt{\frac{1}{8}}$$

Multiply 3 and 2 to get 6.

$$6\sqrt{2}-\frac{1}{2}\sqrt{32}+7\sqrt{\frac{1}{8}}$$

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}.

$$6\sqrt{2}-\frac{1}{2}\times 4\sqrt{2}+7\sqrt{\frac{1}{8}}$$

Express -\frac{1}{2}\times 4 as a single fraction.

$$6\sqrt{2}+\frac{-4}{2}\sqrt{2}+7\sqrt{\frac{1}{8}}$$

Divide -4 by 2 to get -2.

$$6\sqrt{2}-2\sqrt{2}+7\sqrt{\frac{1}{8}}$$

Combine 6\sqrt{2} and -2\sqrt{2} to get 4\sqrt{2}.

$$4\sqrt{2}+7\sqrt{\frac{1}{8}}$$

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

$$4\sqrt{2}+7\times \frac{\sqrt{1}}{\sqrt{8}}$$

Calculate the square root of 1 and get 1.

$$4\sqrt{2}+7\times \frac{1}{\sqrt{8}}$$

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}.

$$4\sqrt{2}+7\times \frac{1}{2\sqrt{2}}$$

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

$$4\sqrt{2}+7\times \frac{\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$4\sqrt{2}+7\times \frac{\sqrt{2}}{2\times 2}$$

Multiply 2 and 2 to get 4.

$$4\sqrt{2}+7\times \frac{\sqrt{2}}{4}$$

Express 7\times \frac{\sqrt{2}}{4} as a single fraction.

$$4\sqrt{2}+\frac{7\sqrt{2}}{4}$$

Combine 4\sqrt{2} and \frac{7\sqrt{2}}{4} to get \frac{23}{4}\sqrt{2}.

$$\frac{23}{4}\sqrt{2}$$