Outcomes
A student:

 MA5.11WM
uses appropriate terminology, diagrams and symbols in mathematical contexts

 MA5.13WM
provides reasoning to support conclusions that are appropriate to the context

 MA5.15NA
operates with algebraic expressions involving positiveinteger and zero indices, and establishes the meaning of negative indices for numerical bases
Content
 Students:
 Extend and apply the index laws to variables, using positiveinteger indices and the zero index (ACMNA212)

use the index laws previously established for numerical bases with positiveinteger indices to develop the index laws in algebraic form, eg
\( \begin{align} 2^2 \times 2^3 &= 2^{2+3} = 2^5 &\text{ leads to } & \,\,\, a^m \times a^n = a^{m+n} \\ 2^5 \div 2^2 &= 2^{52} = 2^3 &\text{ leads to } & \,\,\, a^m \div a^n = a^{mn} \\ (2^2)^3 &= 2^{2 \times 3} = 2^6 &\text{ leads to } & \,\,\, (a^m)^n = a^{mn} \end{align} \)
 explain why a particular algebraic sentence is incorrect, eg explain why \( a^3 \times a^2 = a^6 \) is incorrect (Communicating, Reasoning)

establish that \(x^0 = 1\) using the index laws, eg
\( \begin{align} a^3 \div a^3 &= a^{33} = a^0\\ \textrm{ and }~ a^3 \div a^3 &= 1 \\ \therefore ~ ~ a^0 &= 1 \end{align} \)
 explain why \(x^0 = 1\) (Reasoning)
 simplify expressions that involve the zero index, eg \(5x^0 + 3 = 8\)

simplify expressions that involve the product and quotient of simple algebraic terms containing positiveinteger indices, eg
\( \begin{align} (3x^2)^3 &= 27x^6 \\ 2x^2 \times 3x^3 &= 6x^5 \\ 15a^6 \div 3a^2 &= 5a^4 \\ \frac{3a^2}{15a^6} &= \frac{1}{5a^4} \end{align} \)
 compare expressions such as 3a^{2} × 5a and 3a^{2} + 5a by substituting values for a (Communicating, Reasoning)
 Apply index laws to numerical expressions with integer indices (ACMNA209)

establish the meaning of negative indices for numerical bases, eg by patterns

evaluate numerical expressions involving a negative index by first rewriting with a positive index,
eg \( 3^{4} = \frac{1}{3^4} = \frac{1}{81} \)  write given numbers in index form (integer indices only) and vice versa