Solve (27a^6)^frac{-2{3}}

$$( 27 a ^ { 6 } ) ^ { \frac { - 2 } { 3 } }$$

Evaluate

$$\frac{1}{9a^{4}}$$

Solution Steps

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

$$\left(27a^{6}\right)^{-\frac{2}{3}}$$

Expand \left(27a^{6}\right)^{-\frac{2}{3}}.

$$27^{-\frac{2}{3}}\left(a^{6}\right)^{-\frac{2}{3}}$$

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

$$27^{-\frac{2}{3}}a^{-4}$$

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

$$\frac{1}{9}a^{-4}$$

Differentiate w.r.t. a

$$-\frac{4}{9a^{5}}$$

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).

$$-\frac{2}{3}\times \left(27a^{6}\right)^{-\frac{2}{3}-1}\frac{\mathrm{d}}{\mathrm{d}a}(27a^{6})$$

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}.

$$-\frac{2}{3}\times \left(27a^{6}\right)^{-\frac{5}{3}}\times 6\times 27a^{6-1}$$

Simplify.

$$-108a^{5}\times \left(27a^{6}\right)^{-\frac{5}{3}}$$