Solve {left(8a^3/27x^-3right)}^2/3{left(64a^3/27x^-3right)}^-2/3

$${ \left( \frac{ 8 { a }^{ 3 } }{ 27 } { x }^{ -3 } \right) }^{ } \frac{ 2 }{ 3 } { \left( \frac{ 64 { a }^{ 3 } }{ 27 } { x }^{ -3 } \right) }^{ } \frac{ -2 }{ 3 }$$

Evaluate

$$-\frac{2048\times \left(\frac{a}{x}\right)^{6}}{6561}$$

Solution Steps

Express \frac{8a^{3}}{27}x^{-3} as a single fraction.

$$\left(\frac{8a^{3}x^{-3}}{27}\right)^{1}\times \frac{2}{3}\times \left(\frac{64a^{3}}{27}x^{-3}\right)^{1}\times \frac{-2}{3}$$

Calculate \frac{8a^{3}x^{-3}}{27} to the power of 1 and get \frac{8a^{3}x^{-3}}{27}.

$$\frac{8a^{3}x^{-3}}{27}\times \frac{2}{3}\times \left(\frac{64a^{3}}{27}x^{-3}\right)^{1}\times \frac{-2}{3}$$

Express \frac{64a^{3}}{27}x^{-3} as a single fraction.

$$\frac{8a^{3}x^{-3}}{27}\times \frac{2}{3}\times \left(\frac{64a^{3}x^{-3}}{27}\right)^{1}\times \frac{-2}{3}$$

Calculate \frac{64a^{3}x^{-3}}{27} to the power of 1 and get \frac{64a^{3}x^{-3}}{27}.

$$\frac{8a^{3}x^{-3}}{27}\times \frac{2}{3}\times \frac{64a^{3}x^{-3}}{27}\times \frac{-2}{3}$$

Fraction \frac{-2}{3} can be rewritten as -\frac{2}{3} by extracting the negative sign.

$$\frac{8a^{3}x^{-3}}{27}\times \frac{2}{3}\times \frac{64a^{3}x^{-3}}{27}\left(-\frac{2}{3}\right)$$

Multiply \frac{2}{3} and -\frac{2}{3} to get -\frac{4}{9}.

$$\frac{8a^{3}x^{-3}}{27}\left(-\frac{4}{9}\right)\times \frac{64a^{3}x^{-3}}{27}$$

Multiply \frac{8a^{3}x^{-3}}{27} times -\frac{4}{9} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-8a^{3}x^{-3}\times 4}{27\times 9}\times \frac{64a^{3}x^{-3}}{27}$$

Multiply \frac{-8a^{3}x^{-3}\times 4}{27\times 9} times \frac{64a^{3}x^{-3}}{27} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-8a^{3}x^{-3}\times 4\times 64a^{3}x^{-3}}{27\times 9\times 27}$$

To multiply powers of the same base, add their exponents. Add 3 and 3 to get 6.

$$\frac{-8a^{6}x^{-3}\times 4\times 64x^{-3}}{27\times 9\times 27}$$

To multiply powers of the same base, add their exponents. Add -3 and -3 to get -6.

$$\frac{-8a^{6}x^{-6}\times 4\times 64}{27\times 9\times 27}$$

Multiply -8 and 4 to get -32.

$$\frac{-32a^{6}x^{-6}\times 64}{27\times 9\times 27}$$

Multiply -32 and 64 to get -2048.

$$\frac{-2048a^{6}x^{-6}}{27\times 9\times 27}$$

Multiply 27 and 9 to get 243.

$$\frac{-2048a^{6}x^{-6}}{243\times 27}$$

Multiply 243 and 27 to get 6561.

$$\frac{-2048a^{6}x^{-6}}{6561}$$

Differentiate w.r.t. x

$$\frac{4096a^{6}}{2187x^{7}}$$