Solve (x^{3/4})^6*(16/3)^6:(16/9)^6:(2/3)^6

$$( x ^ { \frac { 3 } { 4 } } ) ^ { 6 } \cdot ( \frac { 16 } { 3 } ) ^ { 6 } : ( \frac { 16 } { 9 } ) ^ { 6 } : ( \frac { 2 } { 3 } ) ^ { 6 }$$

Evaluate

$$\frac{531441x^{\frac{9}{2}}}{64}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply \frac{3}{4} and 6 to get \frac{9}{2}.

$$\frac{\frac{x^{\frac{9}{2}}\times \left(\frac{16}{3}\right)^{6}}{\left(\frac{16}{9}\right)^{6}}}{\left(\frac{2}{3}\right)^{6}}$$

Calculate \frac{16}{3} to the power of 6 and get \frac{16777216}{729}.

$$\frac{\frac{x^{\frac{9}{2}}\times \frac{16777216}{729}}{\left(\frac{16}{9}\right)^{6}}}{\left(\frac{2}{3}\right)^{6}}$$

Calculate \frac{16}{9} to the power of 6 and get \frac{16777216}{531441}.

$$\frac{\frac{x^{\frac{9}{2}}\times \frac{16777216}{729}}{\frac{16777216}{531441}}}{\left(\frac{2}{3}\right)^{6}}$$

Divide x^{\frac{9}{2}}\times \frac{16777216}{729} by \frac{16777216}{531441} by multiplying x^{\frac{9}{2}}\times \frac{16777216}{729} by the reciprocal of \frac{16777216}{531441}.

$$\frac{\frac{x^{\frac{9}{2}}\times \frac{16777216}{729}\times 531441}{16777216}}{\left(\frac{2}{3}\right)^{6}}$$

Multiply \frac{16777216}{729} and 531441 to get 12230590464.

$$\frac{\frac{x^{\frac{9}{2}}\times 12230590464}{16777216}}{\left(\frac{2}{3}\right)^{6}}$$

Divide x^{\frac{9}{2}}\times 12230590464 by 16777216 to get x^{\frac{9}{2}}\times 729.

$$\frac{x^{\frac{9}{2}}\times 729}{\left(\frac{2}{3}\right)^{6}}$$

Calculate \frac{2}{3} to the power of 6 and get \frac{64}{729}.

$$\frac{x^{\frac{9}{2}}\times 729}{\frac{64}{729}}$$

Divide x^{\frac{9}{2}}\times 729 by \frac{64}{729} by multiplying x^{\frac{9}{2}}\times 729 by the reciprocal of \frac{64}{729}.

$$\frac{x^{\frac{9}{2}}\times 729\times 729}{64}$$

Multiply 729 and 729 to get 531441.

$$\frac{x^{\frac{9}{2}}\times 531441}{64}$$

Differentiate w.r.t. x

$$\frac{4782969x^{\frac{7}{2}}}{128}$$