Solve (a^8)^-3*a^-2/a^-6*a

$$\frac { ( a ^ { 8 } ) ^ { - 3 } \times a ^ { - 2 } } { a ^ { - 6 } \times a }$$

Evaluate

$$\frac{1}{a^{21}}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 8 and -3 to get -24.

$$\frac{a^{-24}a^{-2}}{a^{-6}a}$$

To multiply powers of the same base, add their exponents. Add -24 and -2 to get -26.

$$\frac{a^{-26}}{a^{-6}a}$$

To multiply powers of the same base, add their exponents. Add -6 and 1 to get -5.

$$\frac{a^{-26}}{a^{-5}}$$

Rewrite a^{-5} as a^{-26}a^{21}. Cancel out a^{-26} in both numerator and denominator.

$$\frac{1}{a^{21}}$$

Differentiate w.r.t. a

$$-\frac{21}{a^{22}}$$

Steps Using Chain Rule

To raise a power to another power, multiply the exponents. Multiply 8 and -3 to get -24.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{-24}a^{-2}}{a^{-6}a})$$

To multiply powers of the same base, add their exponents. Add -24 and -2 to get -26.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{-26}}{a^{-6}a})$$

To multiply powers of the same base, add their exponents. Add -6 and 1 to get -5.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{-26}}{a^{-5}})$$

Rewrite a^{-5} as a^{-26}a^{21}. Cancel out a^{-26} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{a^{21}})$$

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).

$$-\left(a^{21}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}a}(a^{21})$$

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}.

$$-\left(a^{21}\right)^{-2}\times 21a^{21-1}$$

Simplify.

$$-21a^{20}\left(a^{21}\right)^{-2}$$