Solve a^2*(1/a)^3

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

Evaluate

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

Solution Steps

To raise \frac{1}{a} to a power, raise both numerator and denominator to the power and then divide.

$$a^{2}\times \frac{1^{3}}{a^{3}}$$

Express a^{2}\times \frac{1^{3}}{a^{3}} as a single fraction.

$$\frac{a^{2}\times 1^{3}}{a^{3}}$$

Cancel out a^{2} in both numerator and denominator.

$$\frac{1^{3}}{a}$$

Calculate 1 to the power of 3 and get 1.

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

Differentiate w.r.t. a

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

Steps Using Definition of a Derivative

To raise \frac{1}{a} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{2}\times \frac{1^{3}}{a^{3}})$$

Express a^{2}\times \frac{1^{3}}{a^{3}} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{2}\times 1^{3}}{a^{3}})$$

Cancel out a^{2} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1^{3}}{a})$$

Calculate 1 to the power of 3 and get 1.

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

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

$$-a^{-1-1}$$

Subtract 1 from -1.

$$-a^{-2}$$