Solve frac{1/3sqrt{8}+2/3sqrt{18}}{1/4sqrt{32}}

$$\frac { \frac { 1 } { 3 } \sqrt { 8 } + \frac { 2 } { 3 } \sqrt { 18 } } { \frac { 1 } { 4 } \sqrt { 32 } }$$

Evaluate

$$\frac{8}{3}\approx 2.666666667$$

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}.

$$\frac{\frac{1}{3}\times 2\sqrt{2}+\frac{2}{3}\sqrt{18}}{\frac{1}{4}\sqrt{32}}$$

Multiply \frac{1}{3} and 2 to get \frac{2}{3}.

$$\frac{\frac{2}{3}\sqrt{2}+\frac{2}{3}\sqrt{18}}{\frac{1}{4}\sqrt{32}}$$

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}.

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

Cancel out 3 and 3.

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

Combine \frac{2}{3}\sqrt{2} and 2\sqrt{2} to get \frac{8}{3}\sqrt{2}.

$$\frac{\frac{8}{3}\sqrt{2}}{\frac{1}{4}\sqrt{32}}$$

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}.

$$\frac{\frac{8}{3}\sqrt{2}}{\frac{1}{4}\times 4\sqrt{2}}$$

Cancel out 4 and 4.

$$\frac{\frac{8}{3}\sqrt{2}}{\sqrt{2}}$$

Rationalize the denominator of \frac{\frac{8}{3}\sqrt{2}}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\frac{8}{3}\sqrt{2}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\frac{8}{3}\sqrt{2}\sqrt{2}}{2}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\frac{\frac{8}{3}\times 2}{2}$$

Express \frac{8}{3}\times 2 as a single fraction.

$$\frac{\frac{8\times 2}{3}}{2}$$

Multiply 8 and 2 to get 16.

$$\frac{\frac{16}{3}}{2}$$

Express \frac{\frac{16}{3}}{2} as a single fraction.

$$\frac{16}{3\times 2}$$

Multiply 3 and 2 to get 6.

$$\frac{16}{6}$$

Reduce the fraction \frac{16}{6} to lowest terms by extracting and canceling out 2.

$$\frac{8}{3}$$

Factor

$$\frac{2 ^ {3}}{3} = 2\frac{2}{3} = 2.6666666666666665$$