Solve (2{a}^frac{1{2}})(2{a}^frac{1{3}})(2{a}^frac{1{6}})

$$(2 { a }^{ \frac{ 1 }{ 2 } } )(2 { a }^{ \frac{ 1 }{ 3 } } )(2 { a }^{ \frac{ 1 }{ 6 } } )$$

Evaluate

$$8a$$

Solution Steps

To multiply powers of the same base, add their exponents. Add \frac{1}{2} and \frac{1}{3} to get \frac{5}{6}.

$$2a^{\frac{5}{6}}\times 2\times 2a^{\frac{1}{6}}$$

To multiply powers of the same base, add their exponents. Add \frac{5}{6} and \frac{1}{6} to get 1.

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

Calculate a to the power of 1 and get a.

$$2a\times 2\times 2$$

Multiply 2 and 2 to get 4.

$$4a\times 2$$

Multiply 4 and 2 to get 8.

$$8a$$

Differentiate w.r.t. a

$$8$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add \frac{1}{2} and \frac{1}{3} to get \frac{5}{6}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(2a^{\frac{5}{6}}\times 2\times 2a^{\frac{1}{6}})$$

To multiply powers of the same base, add their exponents. Add \frac{5}{6} and \frac{1}{6} to get 1.

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

Calculate a to the power of 1 and get a.

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

Multiply 2 and 2 to get 4.

$$\frac{\mathrm{d}}{\mathrm{d}a}(4a\times 2)$$

Multiply 4 and 2 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}a}(8a)$$

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

$$8a^{1-1}$$

Subtract 1 from 1.

$$8a^{0}$$

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

$$8\times 1$$

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

$$8$$