Questions: Simplify and convert to radical notation. If r^(5 / 20) = sqrt[-](r^n) then m = and n =

Solution Steps

To solve this problem, we need to simplify the expression $r^{5/20}$ and express it in radical notation. Then, we need to equate it to the given expression $\sqrt[-]{r^{n}}$ to find the values of $m$ and $n$. First, simplify the exponent $5/20$ and express it as a radical. Then, compare the simplified expression to the given form to determine $m$ and $n$.

We start with the expression $r^{5/20}$. Simplifying the exponent gives us: $$\frac{5}{20} = 0.25$$ Thus, we can rewrite the expression as: $$r^{5/20} = r^{0.25}$$

The exponent $0.25$ can be expressed in radical form as: $$r^{0.25} = r^{\frac{1}{4}} = \sqrt[4]{r}$$

We are given that: $$r^{0.25} = \sqrt[-]{r^{n}}$$ This implies: $$\sqrt[4]{r} = r^{n}$$ From our previous simplification, we can equate the exponents: $$n = 0.25$$

The value of $m$ is determined from the simplified exponent $0.25$: $$m = \frac{1}{0.25} = 4$$

Final Answer