Solve (64m^12)^1/2

$$( 64 m ^ { 12 } ) ^ { 1 / 2 }$$

Evaluate

$$8m^{6}$$

Solution Steps

Expand \left(64m^{12}\right)^{\frac{1}{2}}.

$$64^{\frac{1}{2}}\left(m^{12}\right)^{\frac{1}{2}}$$

To raise a power to another power, multiply the exponents. Multiply 12 and \frac{1}{2} to get 6.

$$64^{\frac{1}{2}}m^{6}$$

Calculate 64 to the power of \frac{1}{2} and get 8.

$$8m^{6}$$

Differentiate w.r.t. m

$$48m^{5}$$

Steps Using Chain Rule

If F is the composition of two differentiable functions f\left(u\right) and u=g\left(x\right), that is, if F\left(x\right)=f\left(g\left(x\right)\right), then the derivative of F is the derivative of f with respect to u times the derivative of g with respect to x, that is, \frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right).

$$\frac{1}{2}\times \left(64m^{12}\right)^{\frac{1}{2}-1}\frac{\mathrm{d}}{\mathrm{d}m}(64m^{12})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$\frac{1}{2}\times \left(64m^{12}\right)^{-\frac{1}{2}}\times 12\times 64m^{12-1}$$

Simplify.

$$384m^{11}\times \left(64m^{12}\right)^{-\frac{1}{2}}$$