Solve 10t^-5/t^-2

$$\frac { 10 t ^ { - 5 } } { t ^ { - 2 } }$$

Evaluate

$$\frac{10}{t^{3}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(10t^{-5}\right)^{1}\times \frac{1}{t^{-2}}$$

To raise the product of two or more numbers to a power, raise each number to the power and take their product.

$$10^{1}\left(t^{-5}\right)^{1}\times \frac{1}{1}\times \frac{1}{t^{-2}}$$

Use the Commutative Property of Multiplication.

$$10^{1}\times \frac{1}{1}\left(t^{-5}\right)^{1}\times \frac{1}{t^{-2}}$$

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

$$10^{1}\times \frac{1}{1}t^{-5}t^{-2\left(-1\right)}$$

Multiply -2 times -1.

$$10^{1}\times \frac{1}{1}t^{-5}t^{2}$$

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

$$10^{1}\times \frac{1}{1}t^{-5+2}$$

Add the exponents -5 and 2.

$$10^{1}\times \frac{1}{1}t^{-3}$$

Raise 10 to the power 1.

$$10\times \frac{1}{1}t^{-3}$$

Differentiate w.r.t. t

$$-\frac{30}{t^{4}}$$

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{\mathrm{d}}{\mathrm{d}t}(\frac{10}{1}t^{-5-\left(-2\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}t}(10t^{-3})$$

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}.

$$-3\times 10t^{-3-1}$$

Do the arithmetic.

$$-30t^{-4}$$