Solve 25^-2Xa^-4/5^3*10Xa^-8

$$\frac { 25 ^ { - 2 } X a ^ { - 4 } } { 5 ^ { 3 } \times 10 X a ^ { - 8 } }$$

Evaluate

$$\frac{a^{4}}{781250}$$

Solution Steps

Cancel out X in both numerator and denominator.

$$\frac{25^{-2}a^{-4}}{10\times 5^{3}a^{-8}}$$

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

$$\frac{25^{-2}a^{4}}{10\times 5^{3}}$$

Calculate 25 to the power of -2 and get \frac{1}{625}.

$$\frac{\frac{1}{625}a^{4}}{10\times 5^{3}}$$

Calculate 5 to the power of 3 and get 125.

$$\frac{\frac{1}{625}a^{4}}{10\times 125}$$

Multiply 10 and 125 to get 1250.

$$\frac{\frac{1}{625}a^{4}}{1250}$$

Divide \frac{1}{625}a^{4} by 1250 to get \frac{1}{781250}a^{4}.

$$\frac{1}{781250}a^{4}$$

Differentiate w.r.t. a

$$\frac{2a^{3}}{390625}$$

Steps Using Definition of a Derivative

Cancel out X in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{25^{-2}a^{-4}}{10\times 5^{3}a^{-8}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{25^{-2}a^{4}}{10\times 5^{3}})$$

Calculate 25 to the power of -2 and get \frac{1}{625}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\frac{1}{625}a^{4}}{10\times 5^{3}})$$

Calculate 5 to the power of 3 and get 125.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\frac{1}{625}a^{4}}{10\times 125})$$

Multiply 10 and 125 to get 1250.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\frac{1}{625}a^{4}}{1250})$$

Divide \frac{1}{625}a^{4} by 1250 to get \frac{1}{781250}a^{4}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{781250}a^{4})$$

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

$$4\times \frac{1}{781250}a^{4-1}$$

Multiply 4 times \frac{1}{781250}.

$$\frac{2}{390625}a^{4-1}$$

Subtract 1 from 4.

$$\frac{2}{390625}a^{3}$$