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