Solve (3a^3)(2a^-4)(5a^2)

$$( 3 a ^ { 3 } ) ( 2 a ^ { - 4 } ) ( 5 a ^ { 2 } )$$

Evaluate

$$30a$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 3 and -4 to get -1.

$$3a^{-1}\times 2\times 5a^{2}$$

To multiply powers of the same base, add their exponents. Add -1 and 2 to get 1.

$$3a^{1}\times 2\times 5$$

Calculate a to the power of 1 and get a.

$$3a\times 2\times 5$$

Multiply 3 and 2 to get 6.

$$6a\times 5$$

Multiply 6 and 5 to get 30.

$$30a$$

Differentiate w.r.t. a

$$30$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add 3 and -4 to get -1.

$$\frac{\mathrm{d}}{\mathrm{d}a}(3a^{-1}\times 2\times 5a^{2})$$

To multiply powers of the same base, add their exponents. Add -1 and 2 to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}a}(3a^{1}\times 2\times 5)$$

Calculate a to the power of 1 and get a.

$$\frac{\mathrm{d}}{\mathrm{d}a}(3a\times 2\times 5)$$

Multiply 3 and 2 to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}a}(6a\times 5)$$

Multiply 6 and 5 to get 30.

$$\frac{\mathrm{d}}{\mathrm{d}a}(30a)$$

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

$$30a^{1-1}$$

Subtract 1 from 1.

$$30a^{0}$$

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

$$30\times 1$$

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

$$30$$