Solve (3sqrt{18}+1/5sqrt{50}-4sqrt{1/2})divsqrt{32}

$$( 3 \sqrt { 18 } + \frac { 1 } { 5 } \sqrt { 50 } - 4 \sqrt { \frac { 1 } { 2 } } ) \div \sqrt { 32 }$$

Evaluate

$$2$$

Solution Steps

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

$$\frac{3\times 3\sqrt{2}+\frac{1}{5}\sqrt{50}-4\sqrt{\frac{1}{2}}}{\sqrt{32}}$$

Multiply 3 and 3 to get 9.

$$\frac{9\sqrt{2}+\frac{1}{5}\sqrt{50}-4\sqrt{\frac{1}{2}}}{\sqrt{32}}$$

Factor 50=5^{2}\times 2. Rewrite the square root of the product \sqrt{5^{2}\times 2} as the product of square roots \sqrt{5^{2}}\sqrt{2}. Take the square root of 5^{2}.

$$\frac{9\sqrt{2}+\frac{1}{5}\times 5\sqrt{2}-4\sqrt{\frac{1}{2}}}{\sqrt{32}}$$

Cancel out 5 and 5.

$$\frac{9\sqrt{2}+\sqrt{2}-4\sqrt{\frac{1}{2}}}{\sqrt{32}}$$

Combine 9\sqrt{2} and \sqrt{2} to get 10\sqrt{2}.

$$\frac{10\sqrt{2}-4\sqrt{\frac{1}{2}}}{\sqrt{32}}$$

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

$$\frac{10\sqrt{2}-4\times \frac{\sqrt{1}}{\sqrt{2}}}{\sqrt{32}}$$

Calculate the square root of 1 and get 1.

$$\frac{10\sqrt{2}-4\times \frac{1}{\sqrt{2}}}{\sqrt{32}}$$

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

$$\frac{10\sqrt{2}-4\times \frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}}{\sqrt{32}}$$

The square of \sqrt{2} is 2.

$$\frac{10\sqrt{2}-4\times \frac{\sqrt{2}}{2}}{\sqrt{32}}$$

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

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

Combine 10\sqrt{2} and -2\sqrt{2} to get 8\sqrt{2}.

$$\frac{8\sqrt{2}}{\sqrt{32}}$$

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}.

$$\frac{8\sqrt{2}}{4\sqrt{2}}$$

Cancel out 4\sqrt{2} in both numerator and denominator.

$$2$$

Factor

$$2$$