Sort, with justification, a set of numbers into rational and irrational numbers.

Create and explain a pattern that describes the decimal form of an irrational number (e.g., write the digits from 0 to 9 in order, then put two of each digit – 0011223344... – followed by three of each digit and so on).

Approximate the value of a given irrational number and explain the strategy used.

Order a set of Real numbers, including rational and irrational numbers, on a number line and explain the strategies used.

Express a radical as a mixed radical in simplest form (limited to numerical radicands).

Express a mixed radical as an entire radical (limited to numerical radicands).

Explain, using examples, how changing the value of the index of a radical impacts the value of the radical.

Represent, such as through the use of a graphic organizer, the relationships among the subsets of the Real numbers: natural, whole, integer, rational, and irrational.

Analyze patterns to generalize why $a^{-n}=1/a^n$, $a ≠ 0$

Analyze patterns to generalize why $a^{1/n}=^n√a,n ≠ 0,n∈I$ and $2 > 0$ when $n$ is an even integer.

Extend and apply the exponent laws to powers with rational exponents (limited to expressions with rational and variable bases and integral and rational exponents):

• $(a^m)(a^n)=a^{m+n}$
• $a^m÷a^n=a^{m-n},a ≠ 0$
• $(a^m)^n=a^{mn}$
• $(ab)^m=a^mb^m$
• $(a/b)^n=a^n/b^n,b ≠ 0$

Analyze simplifications of expressions involving radicals and/or powers for errors.

Express powers with rational exponents as radicals and vice versa.

Create a representation that conveys the relationship between powers, rational numbers, and irrational numbers.