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

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

Evaluate

$$4a^{5}$$

Solution Steps

Expand \left(2a\right)^{\frac{1}{2}}.

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

Expand \left(2a^{3}\right)^{\frac{3}{2}}.

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

To raise a power to another power, multiply the exponents. Multiply 3 and \frac{3}{2} to get \frac{9}{2}.

$$2^{\frac{1}{2}}a^{\frac{1}{2}}\times 2^{\frac{3}{2}}a^{\frac{9}{2}}$$

To multiply powers of the same base, add their exponents. Add \frac{1}{2} and \frac{3}{2} to get 2.

$$2^{2}a^{\frac{1}{2}}a^{\frac{9}{2}}$$

To multiply powers of the same base, add their exponents. Add \frac{1}{2} and \frac{9}{2} to get 5.

$$2^{2}a^{5}$$

Calculate 2 to the power of 2 and get 4.

$$4a^{5}$$

Differentiate w.r.t. a

$$20a^{4}$$

Steps Using Definition of a Derivative

Expand \left(2a\right)^{\frac{1}{2}}.

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

Expand \left(2a^{3}\right)^{\frac{3}{2}}.

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

To raise a power to another power, multiply the exponents. Multiply 3 and \frac{3}{2} to get \frac{9}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(2^{\frac{1}{2}}a^{\frac{1}{2}}\times 2^{\frac{3}{2}}a^{\frac{9}{2}})$$

To multiply powers of the same base, add their exponents. Add \frac{1}{2} and \frac{3}{2} to get 2.

$$\frac{\mathrm{d}}{\mathrm{d}a}(2^{2}a^{\frac{1}{2}}a^{\frac{9}{2}})$$

To multiply powers of the same base, add their exponents. Add \frac{1}{2} and \frac{9}{2} to get 5.

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

Calculate 2 to the power of 2 and get 4.

$$\frac{\mathrm{d}}{\mathrm{d}a}(4a^{5})$$

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

$$5\times 4a^{5-1}$$

Multiply 5 times 4.

$$20a^{5-1}$$

Subtract 1 from 5.

$$20a^{4}$$