Indices — NSW Year 10 Advanced Maths
The index laws with algebra, from powers of powers to zero, negative and fractional indices and exponential equations. Five subtopics and 100 practice questions with worked solutions, aligned to NESA outcomes MA5-IND-P-01 and MA5-IND-P-02.
Indices in NSW Year 10 Advanced
Indices are a short way to write repeated multiplication, and the index laws make expressions with powers quick to simplify. In Year 10 Advanced, students apply the laws to algebraic terms, use zero, negative and fractional indices such as \(a^{-n}=\frac{1}{a^n}\) and \(a^{\frac{1}{n}}=\sqrt[n]{a}\), and solve equations with the unknown in the index.
The five subtopics below start with the multiplication and division laws and the power of a power, then cover zero and negative indices and fractional indices, and finish with exponential equations. Indices is one of the 16 topics in NSW Year 10 Advanced Maths and sits within the Number and Algebra strand (NESA outcomes MA5‑IND‑C‑01, MA5‑IND‑P‑01 and MA5‑IND‑P‑02). The index laws lead straight into logarithms, exponential graphs and calculus in Year 11 Advanced.
Subtopics
Indices: common questions
What are the index laws?
For the same base: multiply by adding indices, \(a^m\times a^n=a^{m+n}\); divide by subtracting, \(a^m\div a^n=a^{m-n}\); raise a power to a power by multiplying, \((a^m)^n=a^{mn}\). Also \(a^0=1\) for any \(a\ne0\).
What does a negative index mean?
A negative index means the reciprocal of the positive power: \(a^{-n}=\frac{1}{a^n}\). For example, \(2^{-3}=\frac{1}{8}\) and \(\left(\frac{2}{5}\right)^{-3}=\frac{125}{8}\).
How do you work out a fractional index?
The denominator is a root and the numerator is a power. For example, \(27^{\frac{2}{3}}=\left(\sqrt[3]{27}\right)^2=3^2=9\). Taking the root first keeps the numbers small.
How do you solve an exponential equation?
Write both sides as powers of the same base, then set the indices equal. For \(16^{x-1}=8^{x+2}\): \(2^{4(x-1)}=2^{3(x+2)}\), so \(4x-4=3x+6\) and \(x=10\).
Which NESA outcomes cover Indices?
The index laws with positive and zero indices are MA5‑IND‑C‑01. Negative indices with pronumerals are MA5‑IND‑P‑01, and fractional indices are MA5‑IND‑P‑02, both Path outcomes in Stage 5 of the NSW Mathematics K–10 Syllabus (2022).