Solve sqrt{3sqrt{[(125/729)^2]^4}}=

$$\sqrt { 3 \sqrt { [ ( \frac { 125 } { 729 } ) ^ { 2 } ] ^ { 4 } } } =$$

Evaluate

$$\frac{15625\sqrt{3}}{531441}\approx 0.050924362$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 2 and 4 to get 8.

$$\sqrt{3\sqrt{\left(\frac{125}{729}\right)^{8}}}$$

Calculate \frac{125}{729} to the power of 8 and get \frac{59604644775390625}{79766443076872509863361}.

$$\sqrt{3\sqrt{\frac{59604644775390625}{79766443076872509863361}}}$$

Rewrite the square root of the division \frac{59604644775390625}{79766443076872509863361} as the division of square roots \frac{\sqrt{59604644775390625}}{\sqrt{79766443076872509863361}}. Take the square root of both numerator and denominator.

$$\sqrt{3\times \frac{244140625}{282429536481}}$$

Multiply 3 and \frac{244140625}{282429536481} to get \frac{244140625}{94143178827}.

$$\sqrt{\frac{244140625}{94143178827}}$$

Rewrite the square root of the division \sqrt{\frac{244140625}{94143178827}} as the division of square roots \frac{\sqrt{244140625}}{\sqrt{94143178827}}.

$$\frac{\sqrt{244140625}}{\sqrt{94143178827}}$$

Calculate the square root of 244140625 and get 15625.

$$\frac{15625}{\sqrt{94143178827}}$$

Factor 94143178827=177147^{2}\times 3. Rewrite the square root of the product \sqrt{177147^{2}\times 3} as the product of square roots \sqrt{177147^{2}}\sqrt{3}. Take the square root of 177147^{2}.

$$\frac{15625}{177147\sqrt{3}}$$

Rationalize the denominator of \frac{15625}{177147\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{15625\sqrt{3}}{177147\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{15625\sqrt{3}}{177147\times 3}$$

Multiply 177147 and 3 to get 531441.

$$\frac{15625\sqrt{3}}{531441}$$