Solve (a^8)^-4/aa^4

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

Evaluate

$$\frac{1}{a^{37}}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 8 and -4 to get -32.

$$\frac{a^{-32}}{aa^{4}}$$

To multiply powers of the same base, add their exponents. Add 1 and 4 to get 5.

$$\frac{a^{-32}}{a^{5}}$$

Rewrite a^{5} as a^{-32}a^{37}. Cancel out a^{-32} in both numerator and denominator.

$$\frac{1}{a^{37}}$$

Differentiate w.r.t. a

$$-\frac{37}{a^{38}}$$

Steps Using Chain Rule

To raise a power to another power, multiply the exponents. Multiply 8 and -4 to get -32.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{-32}}{aa^{4}})$$

To multiply powers of the same base, add their exponents. Add 1 and 4 to get 5.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{-32}}{a^{5}})$$

Rewrite a^{5} as a^{-32}a^{37}. Cancel out a^{-32} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{a^{37}})$$

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).

$$-\left(a^{37}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}a}(a^{37})$$

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}.

$$-\left(a^{37}\right)^{-2}\times 37a^{37-1}$$

Simplify.

$$-37a^{36}\left(a^{37}\right)^{-2}$$