Solve a^46*a^-16/a^32

$$\frac { a ^ { 46 } \cdot a ^ { - 16 } } { a ^ { 32 } }$$

Evaluate

$$\frac{1}{a^{2}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 46 and -16 to get 30.

$$\frac{a^{30}}{a^{32}}$$

Rewrite a^{32} as a^{30}a^{2}. Cancel out a^{30} in both numerator and denominator.

$$\frac{1}{a^{2}}$$

Differentiate w.r.t. a

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

Steps Using Chain Rule

To multiply powers of the same base, add their exponents. Add 46 and -16 to get 30.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{30}}{a^{32}})$$

Rewrite a^{32} as a^{30}a^{2}. Cancel out a^{30} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{a^{2}})$$

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).

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

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}.

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

Simplify.

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

For any term t, t^{1}=t.

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