Indices and Scientific Notation — NSW Year 9 Advanced Maths
Index laws for zero, negative and fractional indices, plus scientific notation. Nine subtopics and 180 questions with worked solutions, aligned to NESA outcomes MA5-IND-C-01, MA5-IND-P-01, MA5-IND-P-02 and MA5-MAG-C-01.
Indices and Scientific Notation in NSW Year 9 Advanced
Index notation writes repeated multiplication compactly, and the index laws turn long calculations into quick ones. In Year 9 Advanced, students apply the laws to algebraic terms, give meaning to zero, negative and fractional indices, and use powers of 10 to write very large and very small measurements in scientific notation, rounded to a sensible number of significant figures.
The nine subtopics below cover the index laws one at a time, then negative and fractional indices and mixed questions, and finish with scientific notation and significant figures. Indices and Scientific Notation is one of the 14 topics in NSW Year 9 Advanced Maths. The index work belongs to NESA's Number and Algebra strand (NESA outcomes MA5‑IND‑C‑01, MA5‑IND‑P‑01 and MA5‑IND‑P‑02), and scientific notation is MA5‑MAG‑C‑01. Indices lead on to exponential graphs and, later, logarithms.
Subtopics
Indices and Scientific Notation: common questions
What are the index laws?
For the same base, \(a^m \times a^n = a^{m+n}\), \(a^m \div a^n = a^{m-n}\) and \((a^m)^n = a^{mn}\), and \(a^0 = 1\) for any non-zero \(a\). For example, \(\frac{48a^7b^5}{16a^5b^2} = 3a^2b^3\).
What does a negative index mean?
A negative index means a reciprocal: \(a^{-n} = \frac{1}{a^n}\). So \(2^{-1} + 3^{-2} = \frac{1}{2} + \frac{1}{9} = \frac{11}{18}\), and \((3m^4)^{-2} = \frac{1}{9m^8}\).
What does a fractional index mean?
The denominator gives a root and the numerator gives a power: \(a^{\frac{m}{n}} = (\sqrt[n]{a})^m\). For example, \(27^{\frac{2}{3}} = (\sqrt[3]{27})^2 = 3^2 = 9\).
How do you write a number in scientific notation?
Write it as a number from 1 up to (but not including) 10, multiplied by a power of 10. So \(0.000000045 = 4.5 \times 10^{-8}\) and \(708\,000 = 7.08 \times 10^5\).
Which NESA outcomes cover indices in Year 9 Advanced?
Positive and zero indices are the Stage 5 Core outcome MA5‑IND‑C‑01. Negative indices with algebraic bases are MA5‑IND‑P‑01 and fractional indices are MA5‑IND‑P‑02, both Path outcomes. Scientific notation and significant figures are MA5‑MAG‑C‑01.