Solve 25*t^-8/5^-3*10*t^-9

$$\frac { 25 \times t ^ { - 8 } } { 5 ^ { - 3 } \times 10 \times t ^ { - 9 } }$$

Evaluate

$$\frac{625t}{2}$$

Solution Steps

Cancel out 5 in both numerator and denominator.

$$\frac{5t^{-8}}{2\times 5^{-3}t^{-9}}$$

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

$$\frac{5^{4}t^{1}}{2}$$

Calculate 5 to the power of 4 and get 625.

$$\frac{625t^{1}}{2}$$

Calculate t to the power of 1 and get t.

$$\frac{625t}{2}$$

Differentiate w.r.t. t

$$\frac{625}{2} = 312\frac{1}{2} = 312.5$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}t}(\frac{25}{\frac{2}{25}}t^{-8-\left(-9\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}t}(\frac{625}{2}t^{1})$$

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}.

$$\frac{625}{2}t^{1-1}$$

Do the arithmetic.

$$\frac{625}{2}t^{0}$$

For any term t except 0, t^{0}=1.

$$\frac{625}{2}\times 1$$

For any term t, t\times 1=t and 1t=t.

$$\frac{625}{2}$$