NSW Syllabuses

# Mathematics K–10 - Stage 5.1 - Number and Algebra Indices

## Outcomes

#### A student:

• MA5.1-1WM

uses appropriate terminology, diagrams and symbols in mathematical contexts

• MA5.1-3WM

provides reasoning to support conclusions that are appropriate to the context

• MA5.1-5NA

operates with algebraic expressions involving positive-integer and zero indices, and establishes the meaning of negative indices for numerical bases

## Content

• use the index laws previously established for numerical bases with positive-integer 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^{5-2} = 2^3 &\text{ leads to } & \,\,\, a^m \div a^n = a^{m-n} \\ (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^{3-3} = 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 positive-integer 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 $$3a\times5a$$ 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