Solve (81r^2)^frac{1{2}}

$$( 81 r ^ { 2 } ) ^ { \frac { 1 } { 2 } }$$

Evaluate

$$9|r|$$

Solution Steps

Expand \left(81r^{2}\right)^{\frac{1}{2}}.

$$81^{\frac{1}{2}}\left(r^{2}\right)^{\frac{1}{2}}$$

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

$$81^{\frac{1}{2}}r^{1}$$

Calculate 81 to the power of \frac{1}{2} and get 9.

$$9r^{1}$$

Calculate r to the power of 1 and get r.

$$9r$$

Differentiate w.r.t. r

$$9sign(r)$$

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).

$$\frac{1}{2}\times \left(81r^{2}\right)^{\frac{1}{2}-1}\frac{\mathrm{d}}{\mathrm{d}r}(81r^{2})$$

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}.

$$\frac{1}{2}\times \left(81r^{2}\right)^{-\frac{1}{2}}\times 2\times 81r^{2-1}$$

Simplify.

$$81r^{1}\times \left(81r^{2}\right)^{-\frac{1}{2}}$$

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

$$81r\times \left(81r^{2}\right)^{-\frac{1}{2}}$$