Solve 5/3sqrt{8}:sqrt{17/18}

$$\frac { 5 } { 3 } \sqrt { 8 } : \sqrt { 1 \frac { 7 } { 18 } }$$

Evaluate

$$4$$

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}.

$$\frac{\frac{5}{3}\times 2\sqrt{2}}{\sqrt{\frac{1\times 18+7}{18}}}$$

Express \frac{5}{3}\times 2 as a single fraction.

$$\frac{\frac{5\times 2}{3}\sqrt{2}}{\sqrt{\frac{1\times 18+7}{18}}}$$

Multiply 5 and 2 to get 10.

$$\frac{\frac{10}{3}\sqrt{2}}{\sqrt{\frac{1\times 18+7}{18}}}$$

Multiply 1 and 18 to get 18.

$$\frac{\frac{10}{3}\sqrt{2}}{\sqrt{\frac{18+7}{18}}}$$

Add 18 and 7 to get 25.

$$\frac{\frac{10}{3}\sqrt{2}}{\sqrt{\frac{25}{18}}}$$

Rewrite the square root of the division \sqrt{\frac{25}{18}} as the division of square roots \frac{\sqrt{25}}{\sqrt{18}}.

$$\frac{\frac{10}{3}\sqrt{2}}{\frac{\sqrt{25}}{\sqrt{18}}}$$

Calculate the square root of 25 and get 5.

$$\frac{\frac{10}{3}\sqrt{2}}{\frac{5}{\sqrt{18}}}$$

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}.

$$\frac{\frac{10}{3}\sqrt{2}}{\frac{5}{3\sqrt{2}}}$$

Rationalize the denominator of \frac{5}{3\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\frac{10}{3}\sqrt{2}}{\frac{5\sqrt{2}}{3\left(\sqrt{2}\right)^{2}}}$$

The square of \sqrt{2} is 2.

$$\frac{\frac{10}{3}\sqrt{2}}{\frac{5\sqrt{2}}{3\times 2}}$$

Multiply 3 and 2 to get 6.

$$\frac{\frac{10}{3}\sqrt{2}}{\frac{5\sqrt{2}}{6}}$$

Divide \frac{10}{3}\sqrt{2} by \frac{5\sqrt{2}}{6} by multiplying \frac{10}{3}\sqrt{2} by the reciprocal of \frac{5\sqrt{2}}{6}.

$$\frac{\frac{10}{3}\sqrt{2}\times 6}{5\sqrt{2}}$$

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

$$\frac{\frac{10}{3}\times 6}{5}$$

Express \frac{10}{3}\times 6 as a single fraction.

$$\frac{\frac{10\times 6}{3}}{5}$$

Multiply 10 and 6 to get 60.

$$\frac{\frac{60}{3}}{5}$$

Divide 60 by 3 to get 20.

$$\frac{20}{5}$$

Divide 20 by 5 to get 4.

$$4$$

Factor

$$2^{2}$$