Solve (25^3)^-frac{1{2}}a

$$( 25 ^ { 3 } ) ^ { - \frac { 1 } { 2 } } a$$

Evaluate

$$\frac{a}{125}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 3 and -\frac{1}{2} to get -\frac{3}{2}.

$$25^{-\frac{3}{2}}a$$

Calculate 25 to the power of -\frac{3}{2} and get \frac{1}{125}.

$$\frac{1}{125}a$$

Differentiate w.r.t. a

$$\frac{1}{125} = 0.008$$

Steps Using Definition of a Derivative

To raise a power to another power, multiply the exponents. Multiply 3 and -\frac{1}{2} to get -\frac{3}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(25^{-\frac{3}{2}}a)$$

Calculate 25 to the power of -\frac{3}{2} and get \frac{1}{125}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{125}a)$$

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

$$\frac{1}{125}a^{1-1}$$

Subtract 1 from 1.

$$\frac{1}{125}a^{0}$$

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

$$\frac{1}{125}\times 1$$

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

$$\frac{1}{125}$$