Solve (4b)^1/2*(8b^14)

$$( 4 b ) ^ { 1 / 2 } \cdot ( 8 b ^ { 14 } )$$

Evaluate

$$16b^{\frac{29}{2}}$$

Solution Steps

Expand \left(4b\right)^{\frac{1}{2}}.

$$4^{\frac{1}{2}}b^{\frac{1}{2}}\times 8b^{14}$$

Calculate 4 to the power of \frac{1}{2} and get 2.

$$2b^{\frac{1}{2}}\times 8b^{14}$$

Multiply 2 and 8 to get 16.

$$16b^{\frac{1}{2}}b^{14}$$

To multiply powers of the same base, add their exponents. Add \frac{1}{2} and 14 to get \frac{29}{2}.

$$16b^{\frac{29}{2}}$$

Differentiate w.r.t. b

$$232b^{\frac{27}{2}}$$

Steps Using Definition of a Derivative

Expand \left(4b\right)^{\frac{1}{2}}.

$$\frac{\mathrm{d}}{\mathrm{d}b}(4^{\frac{1}{2}}b^{\frac{1}{2}}\times 8b^{14})$$

Calculate 4 to the power of \frac{1}{2} and get 2.

$$\frac{\mathrm{d}}{\mathrm{d}b}(2b^{\frac{1}{2}}\times 8b^{14})$$

Multiply 2 and 8 to get 16.

$$\frac{\mathrm{d}}{\mathrm{d}b}(16b^{\frac{1}{2}}b^{14})$$

To multiply powers of the same base, add their exponents. Add \frac{1}{2} and 14 to get \frac{29}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}b}(16b^{\frac{29}{2}})$$

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

$$\frac{29}{2}\times 16b^{\frac{29}{2}-1}$$

Multiply \frac{29}{2} times 16.

$$232b^{\frac{29}{2}-1}$$

Subtract 1 from \frac{29}{2}.

$$232b^{\frac{27}{2}}$$