Students will learn how to determine whether an expression with n-th roots is equivalent to an expression with a rational power. First, students will learn how to do this for unit fraction exponents. Here's a lesson, and here's practice. Then students will learn how to do this for non-unit fraction exponents. Here's a lesson on that. Students can practice here and here.

Conclude by giving your students these challenges:

- Song Book by NRICH
- Palindromic Date by NRICH
- 2008 AMC 8, Problem 25

Let \(a,\) \(b,\) \(c,\) \(d,\) \(e,\) \(f,\) denote non-zero decimal digits. Two \(3\)-digit natural numbers \(abc\) and \(def\) satisfy \(abc + def = 1000.\) Find the sum \(a + b + c + d + e + f.\) Here's the answer:

We must have

$$\begin{align} & c + f = 10 \\ & 1 + b + e = 10 \\ & 1 + a + d = 10 \end{align}$$Adding all the equations together gives us

$$a + b + c + d + e + f + 2 = 30$$Then subtracting \(2\) from both sides gives us our answer

$$a + b + c + d + e + f = 28$$Now let's extend the problem. Let \(a_1,\) \(\ldots,\) \(a_n,\) \(b_1,\) \(\ldots,\) \(b_n,\) denote non-zero decimal digits. Two \(n\)-digit natural numbers \(a_1a_2\ldots a_n\) and \(b_1b_2\ldots b_n\) satisfy \(a_1a_2\ldots a_n + b_1b_2\ldots b_n = 10^n.\) Find the sum \(a_1 + \ldots + a_n + b_1 + \ldots + b_n.\) Here's the answer:

We must have

$$\begin{align} & a_n + b_n = 10 \\ & 1 + a_{n - 1} + b_{n - 1} = 10 \\ & \ldots \\ & 1 + a_1 + b_1 = 10 \end{align}$$Adding all the equations together gives us

$$a_1 + \ldots + a_n + b_1 + \ldots + b_n + (n - 1) = 10n$$Then subtracting \(n - 1\) from both sides gives us our answer

$$a_1 + \ldots + a_n + b_1 + \ldots + b_n = 9n + 1$$