IP EMaths Notes (Upper Sec, Year 3-4): 02) Indices and Standard Form

Scientific notation keeps working memory clear when comparing magnitudes or evaluating calculator steps. Pair it with consistent index laws.

Essential identities

• \(a^{m} \times a^{n} = a^{m+n}\) and \(\dfrac{a^{m}}{a^{n}} = a^{m-n}\) for \(a \neq 0\).
• \(\bigl(a^{m}\bigr)^{n} = a^{mn}\) and \(a^{-n} = \dfrac{1}{a^{n}}\).
• Standard form: \(N = k \times 10^{n}\) with \(1 \leq k < 10\) and \(n\) an integer.

Worked example - Multiply and compare magnitudes

Compute \((4.2 \times 10^{5})(3 \times 10^{-3})\) and express the answer in standard form.

1. Multiply the decimal parts: \(4.2 \times 3 = 12.6\).
2. Add the indices: \(10^{5} \times 10^{-3} = 10^{2}\).
3. Combine: \(12.6 \times 10^{2}\).
4. Adjust to standard form: \(12.6 = 1.26 \times 10^{1}\), so \(12.6 \times 10^{2} = 1.26 \times 10^{3}\).

Hence the product is \(1.26 \times 10^{3}\).

Try this

Write \(\dfrac{0.00084}{2.1 \times 10^{-4}}\) in standard form, showing every index step.