Solve a^4*(a^-2)^3diva

$$a ^ { 4 } \times ( a ^ { - 2 } ) ^ { 3 } \div a$$

Evaluate

$$\frac{1}{a^{3}}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply -2 and 3 to get -6.

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

To multiply powers of the same base, add their exponents. Add 4 and -6 to get -2.

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

Rewrite a as a^{-2}a^{3}. Cancel out a^{-2} in both numerator and denominator.

$$\frac{1}{a^{3}}$$

Differentiate w.r.t. a

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

Steps Using Chain Rule

To raise a power to another power, multiply the exponents. Multiply -2 and 3 to get -6.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{4}a^{-6}}{a})$$

To multiply powers of the same base, add their exponents. Add 4 and -6 to get -2.

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

Rewrite a as a^{-2}a^{3}. Cancel out a^{-2} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{a^{3}})$$

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).

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

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}.

$$-\left(a^{3}\right)^{-2}\times 3a^{3-1}$$

Simplify.

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