Solve C15^3*5^5/5^2*5^0

$$C 1 \frac { 5 ^ { 3 } \cdot 5 ^ { 5 } } { 5 ^ { 2 } \cdot 5 ^ { 0 } }$$

Evaluate

$$15625C_{1}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 3 and 5 to get 8.

$$C_{1}\times \frac{5^{8}}{5^{2}\times 5^{0}}$$

To multiply powers of the same base, add their exponents. Add 2 and 0 to get 2.

$$C_{1}\times \frac{5^{8}}{5^{2}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 2 from 8 to get 6.

$$C_{1}\times 5^{6}$$

Calculate 5 to the power of 6 and get 15625.

$$C_{1}\times 15625$$

Differentiate w.r.t. C_1

$$15625$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add 3 and 5 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}C_{1}}(C_{1}\times \frac{5^{8}}{5^{2}\times 5^{0}})$$

To multiply powers of the same base, add their exponents. Add 2 and 0 to get 2.

$$\frac{\mathrm{d}}{\mathrm{d}C_{1}}(C_{1}\times \frac{5^{8}}{5^{2}})$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 2 from 8 to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}C_{1}}(C_{1}\times 5^{6})$$

Calculate 5 to the power of 6 and get 15625.

$$\frac{\mathrm{d}}{\mathrm{d}C_{1}}(C_{1}\times 15625)$$

The derivative of ax^{n} is nax^{n-1}.

$$15625C_{1}^{1-1}$$

Subtract 1 from 1.

$$15625C_{1}^{0}$$

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

$$15625\times 1$$

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

$$15625$$