Solve sqrt{3^-1+6^-1/9^-1}

$$\sqrt { \frac { 3 ^ { - 1 } + 6 ^ { - 1 } } { 9 ^ { - 1 } } }$$

Evaluate

$$\frac{3\sqrt{2}}{2}\approx 2.121320344$$

Solution Steps

Calculate 3 to the power of -1 and get \frac{1}{3}.

$$\sqrt{\frac{\frac{1}{3}+6^{-1}}{9^{-1}}}$$

Calculate 6 to the power of -1 and get \frac{1}{6}.

$$\sqrt{\frac{\frac{1}{3}+\frac{1}{6}}{9^{-1}}}$$

Add \frac{1}{3} and \frac{1}{6} to get \frac{1}{2}.

$$\sqrt{\frac{\frac{1}{2}}{9^{-1}}}$$

Calculate 9 to the power of -1 and get \frac{1}{9}.

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

Divide \frac{1}{2} by \frac{1}{9} by multiplying \frac{1}{2} by the reciprocal of \frac{1}{9}.

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

Multiply \frac{1}{2} and 9 to get \frac{9}{2}.

$$\sqrt{\frac{9}{2}}$$

Rewrite the square root of the division \sqrt{\frac{9}{2}} as the division of square roots \frac{\sqrt{9}}{\sqrt{2}}.

$$\frac{\sqrt{9}}{\sqrt{2}}$$

Calculate the square root of 9 and get 3.

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

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

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

The square of \sqrt{2} is 2.

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