Solve (t^-2)^4/t^-1

$$\frac { ( t ^ { - 2 } ) ^ { 4 } } { t ^ { - 1 } }$$

Evaluate

$$\frac{1}{t^{7}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(t^{-2}\right)^{4}\times \frac{1}{\frac{1}{t}}$$

To raise a power to another power, multiply the exponents.

$$t^{-2\times 4}t^{-\left(-1\right)}$$

Multiply -2 times 4.

$$t^{-8}t^{-\left(-1\right)}$$

Multiply -1 times -1.

$$t^{-8}t^{1}$$

To multiply powers of the same base, add their exponents.

$$t^{-8+1}$$

Add the exponents -8 and 1.

$$t^{-7}$$

Differentiate w.r.t. t

$$-\frac{7}{t^{8}}$$

Steps Using Chain Rule

To raise a power to another power, multiply the exponents. Multiply -2 and 4 to get -8.

$$\frac{\mathrm{d}}{\mathrm{d}t}(\frac{t^{-8}}{t^{-1}})$$

Rewrite t^{-1} as t^{-8}t^{7}. Cancel out t^{-8} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}t}(\frac{1}{t^{7}})$$

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).

$$-\left(t^{7}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}t}(t^{7})$$

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}.

$$-\left(t^{7}\right)^{-2}\times 7t^{7-1}$$

Simplify.

$$-7t^{6}\left(t^{7}\right)^{-2}$$