Solve y:(5^20div5^5)*5^3

$$y : ( 5 ^ { 20 } \div 5 ^ { 5 } ) \times 5 ^ { 3 }$$

Evaluate

$$\frac{y}{244140625}$$

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 5 from 20 to get 15.

$$\frac{y}{5^{15}}\times 5^{3}$$

Calculate 5 to the power of 15 and get 30517578125.

$$\frac{y}{30517578125}\times 5^{3}$$

Calculate 5 to the power of 3 and get 125.

$$\frac{y}{30517578125}\times 125$$

Cancel out 30517578125, the greatest common factor in 125 and 30517578125.

$$\frac{y}{244140625}$$

Differentiate w.r.t. y

$$\frac{1}{244140625} = 4.096 \times 10^{-9}$$

Steps Using Definition of a Derivative

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 5 from 20 to get 15.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{y}{5^{15}}\times 5^{3})$$

Calculate 5 to the power of 15 and get 30517578125.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{y}{30517578125}\times 5^{3})$$

Calculate 5 to the power of 3 and get 125.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{y}{30517578125}\times 125)$$

Cancel out 30517578125, the greatest common factor in 125 and 30517578125.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{y}{244140625})$$

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

$$\frac{1}{244140625}y^{1-1}$$

Subtract 1 from 1.

$$\frac{1}{244140625}y^{0}$$

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

$$\frac{1}{244140625}\times 1$$

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

$$\frac{1}{244140625}$$