Solve 1/3/sqrt{frac{8{9}}}*sqrt{frac{8/9}}{sqrt{8/9}}=

$$\frac { \frac { 1 } { 3 } } { \sqrt { \frac { 8 } { 9 } } } \times \frac { \sqrt { \frac { 8 } { 9 } } } { \sqrt { \frac { 8 } { 9 } } } =$$

Evaluate

$$\frac{\sqrt{2}}{4}\approx 0.353553391$$

Solution Steps

Divide \sqrt{\frac{8}{9}} by \sqrt{\frac{8}{9}} to get 1.

$$\frac{\frac{1}{3}}{\sqrt{\frac{8}{9}}}\times 1$$

Rewrite the square root of the division \sqrt{\frac{8}{9}} as the division of square roots \frac{\sqrt{8}}{\sqrt{9}}.

$$\frac{\frac{1}{3}}{\frac{\sqrt{8}}{\sqrt{9}}}\times 1$$

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}.

$$\frac{\frac{1}{3}}{\frac{2\sqrt{2}}{\sqrt{9}}}\times 1$$

Calculate the square root of 9 and get 3.

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

Divide \frac{1}{3} by \frac{2\sqrt{2}}{3} by multiplying \frac{1}{3} by the reciprocal of \frac{2\sqrt{2}}{3}.

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

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

$$\frac{3\sqrt{2}}{3\times 2\left(\sqrt{2}\right)^{2}}\times 1$$

The square of \sqrt{2} is 2.

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

Cancel out 3 in both numerator and denominator.

$$\frac{\sqrt{2}}{2\times 2}\times 1$$

Multiply 2 and 2 to get 4.

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

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

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