Solve (121m^8)^1/2

$$( 121 m ^ { 8 } ) ^ { 1 / 2 }$$

Evaluate

$$11m^{4}$$

Solution Steps

Expand \left(121m^{8}\right)^{\frac{1}{2}}.

$$121^{\frac{1}{2}}\left(m^{8}\right)^{\frac{1}{2}}$$

To raise a power to another power, multiply the exponents. Multiply 8 and \frac{1}{2} to get 4.

$$121^{\frac{1}{2}}m^{4}$$

Calculate 121 to the power of \frac{1}{2} and get 11.

$$11m^{4}$$

Differentiate w.r.t. m

$$44m^{3}$$

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).

$$\frac{1}{2}\times \left(121m^{8}\right)^{\frac{1}{2}-1}\frac{\mathrm{d}}{\mathrm{d}m}(121m^{8})$$

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}.

$$\frac{1}{2}\times \left(121m^{8}\right)^{-\frac{1}{2}}\times 8\times 121m^{8-1}$$

Simplify.

$$484m^{7}\times \left(121m^{8}\right)^{-\frac{1}{2}}$$