Solve 3^-2*813/4*(1/9)^-1/sqrt[4]{9*sqrt[3]{9^-frac{3{2}}}*sqrt{27}}

$$\frac { 3 ^ { - 2 } \cdot 81 \frac { 3 } { 4 } \cdot ( \frac { 1 } { 9 } ) ^ { - 1 } } { \sqrt[ 4 ] { 9 } \cdot \sqrt[ 3 ] { 9 ^ { - \frac { 3 } { 2 } } } \cdot \sqrt { 27 } }$$

Evaluate

$$\frac{109}{4}=27.25$$

Solution Steps

Calculate 3 to the power of -2 and get \frac{1}{9}.

$$\frac{\frac{1}{9}\times \frac{81\times 4+3}{4}\times \left(\frac{1}{9}\right)^{-1}}{\sqrt[4]{9}\sqrt[3]{9^{-\frac{3}{2}}}\sqrt{27}}$$

Multiply 81 and 4 to get 324.

$$\frac{\frac{1}{9}\times \frac{324+3}{4}\times \left(\frac{1}{9}\right)^{-1}}{\sqrt[4]{9}\sqrt[3]{9^{-\frac{3}{2}}}\sqrt{27}}$$

Add 324 and 3 to get 327.

$$\frac{\frac{1}{9}\times \frac{327}{4}\times \left(\frac{1}{9}\right)^{-1}}{\sqrt[4]{9}\sqrt[3]{9^{-\frac{3}{2}}}\sqrt{27}}$$

Multiply \frac{1}{9} and \frac{327}{4} to get \frac{109}{12}.

$$\frac{\frac{109}{12}\times \left(\frac{1}{9}\right)^{-1}}{\sqrt[4]{9}\sqrt[3]{9^{-\frac{3}{2}}}\sqrt{27}}$$

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

$$\frac{\frac{109}{12}\times 9}{\sqrt[4]{9}\sqrt[3]{9^{-\frac{3}{2}}}\sqrt{27}}$$

Multiply \frac{109}{12} and 9 to get \frac{327}{4}.

$$\frac{\frac{327}{4}}{\sqrt[4]{9}\sqrt[3]{9^{-\frac{3}{2}}}\sqrt{27}}$$

Rewrite \sqrt[4]{9} as \sqrt[4]{3^{2}}. Convert from radical to exponential form and cancel out 2 in the exponent. Convert back to radical form.

$$\sqrt[4]{9}=\sqrt[4]{3^{2}}=3^{\frac{2}{4}}=3^{\frac{1}{2}}=\sqrt{3}$$

Insert the obtained value back in the expression.

$$\frac{\frac{327}{4}}{\sqrt{3}\sqrt[3]{9^{-\frac{3}{2}}}\sqrt{27}}$$

Calculate 9 to the power of -\frac{3}{2} and get \frac{1}{27}.

$$\frac{\frac{327}{4}}{\sqrt{3}\sqrt[3]{\frac{1}{27}}\sqrt{27}}$$

Calculate \sqrt[3]{\frac{1}{27}} and get \frac{1}{3}.

$$\frac{\frac{327}{4}}{\sqrt{3}\times \frac{1}{3}\sqrt{27}}$$

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

$$\frac{\frac{327}{4}}{\sqrt{3}\times \frac{1}{3}\times 3\sqrt{3}}$$

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

$$\frac{\frac{327}{4}}{\sqrt{3}\sqrt{3}}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{\frac{327}{4}}{3}$$

Express \frac{\frac{327}{4}}{3} as a single fraction.

$$\frac{327}{4\times 3}$$

Multiply 4 and 3 to get 12.

$$\frac{327}{12}$$

Reduce the fraction \frac{327}{12} to lowest terms by extracting and canceling out 3.

$$\frac{109}{4}$$

Factor

$$\frac{109}{2 ^ {2}} = 27\frac{1}{4} = 27.25$$