Solve (81a^-8)^-frac{3{4}}

$$( 81 a ^ { - 8 } ) ^ { - \frac { 3 } { 4 } }$$

Evaluate

$$\frac{a^{6}}{27}$$

Solution Steps

Expand \left(81a^{-8}\right)^{-\frac{3}{4}}.

$$81^{-\frac{3}{4}}\left(a^{-8}\right)^{-\frac{3}{4}}$$

To raise a power to another power, multiply the exponents. Multiply -8 and -\frac{3}{4} to get 6.

$$81^{-\frac{3}{4}}a^{6}$$

Calculate 81 to the power of -\frac{3}{4} and get \frac{1}{27}.

$$\frac{1}{27}a^{6}$$

Differentiate w.r.t. a

$$\frac{2a^{5}}{9}$$

Steps Using Chain Rule

If F is the composition of two differentiable functions f\left(u\right) and u=g\left(x\right), that is, if F\left(x\right)=f\left(g\left(x\right)\right), then the derivative of F is the derivative of f with respect to u times the derivative of g with respect to x, that is, \frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right).

$$-\frac{3}{4}\times \left(81a^{-8}\right)^{-\frac{3}{4}-1}\frac{\mathrm{d}}{\mathrm{d}a}(81a^{-8})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$-\frac{3}{4}\times \left(81a^{-8}\right)^{-\frac{7}{4}}\left(-8\right)\times 81a^{-8-1}$$

Simplify.

$$486a^{-9}\times \left(81a^{-8}\right)^{-\frac{7}{4}}$$