Solve 3sqrt{42}-2sqrt{1/8}+2sqrt{1/32}

$$3 \sqrt { 42 } - 2 \sqrt { \frac { 1 } { 8 } } + 2 \sqrt { \frac { 1 } { 32 } }$$

Evaluate

$$-\frac{\sqrt{2}}{4}+3\sqrt{42}\approx 19.088668705$$

Solution Steps

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

$$3\sqrt{42}-2\times \frac{\sqrt{1}}{\sqrt{8}}+2\sqrt{\frac{1}{32}}$$

Calculate the square root of 1 and get 1.

$$3\sqrt{42}-2\times \frac{1}{\sqrt{8}}+2\sqrt{\frac{1}{32}}$$

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}.

$$3\sqrt{42}-2\times \frac{1}{2\sqrt{2}}+2\sqrt{\frac{1}{32}}$$

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

$$3\sqrt{42}-2\times \frac{\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}+2\sqrt{\frac{1}{32}}$$

The square of \sqrt{2} is 2.

$$3\sqrt{42}-2\times \frac{\sqrt{2}}{2\times 2}+2\sqrt{\frac{1}{32}}$$

Multiply 2 and 2 to get 4.

$$3\sqrt{42}-2\times \frac{\sqrt{2}}{4}+2\sqrt{\frac{1}{32}}$$

Cancel out 4, the greatest common factor in 2 and 4.

$$3\sqrt{42}+\frac{\sqrt{2}}{-2}+2\sqrt{\frac{1}{32}}$$

Rewrite the square root of the division \sqrt{\frac{1}{32}} as the division of square roots \frac{\sqrt{1}}{\sqrt{32}}.

$$3\sqrt{42}+\frac{\sqrt{2}}{-2}+2\times \frac{\sqrt{1}}{\sqrt{32}}$$

Calculate the square root of 1 and get 1.

$$3\sqrt{42}+\frac{\sqrt{2}}{-2}+2\times \frac{1}{\sqrt{32}}$$

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}.

$$3\sqrt{42}+\frac{\sqrt{2}}{-2}+2\times \frac{1}{4\sqrt{2}}$$

Rationalize the denominator of \frac{1}{4\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$3\sqrt{42}+\frac{\sqrt{2}}{-2}+2\times \frac{\sqrt{2}}{4\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$3\sqrt{42}+\frac{\sqrt{2}}{-2}+2\times \frac{\sqrt{2}}{4\times 2}$$

Multiply 4 and 2 to get 8.

$$3\sqrt{42}+\frac{\sqrt{2}}{-2}+2\times \frac{\sqrt{2}}{8}$$

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

$$3\sqrt{42}+\frac{\sqrt{2}}{-2}+\frac{\sqrt{2}}{4}$$

Combine \frac{\sqrt{2}}{-2} and \frac{\sqrt{2}}{4} to get -\frac{1}{4}\sqrt{2}.

$$3\sqrt{42}-\frac{1}{4}\sqrt{2}$$