Indices and scientific notation

What you will learn

• 10.1 Laws of positive integral indices.
• 10.2 Zero and negative indices: a⁰ = 1, a⁻ⁿ = 1/aⁿ (a ≠ 0).
• 10.3 The same laws work for all integer indices.
• 10.4 Scientific notation: a × 10ⁿ, where 1 ≤ a < 10 and n is an integer.
• NF 10.5 Binary and denary systems; converting between them.
• Hexadecimal is enrichment only (removed from the senior curriculum).

You need first: Number and algebra basics

Key terms

Standard steps

The five laws

a⁰ = 1, because aᵖ ÷ aᵖ = 1. a⁻ⁿ = 1/aⁿ, because a⁰ ÷ aⁿ = 1/aⁿ.

1. Write the terms with the same base where you can (8 = 2³).
2. Pick the law for each operation: × adds indices, ÷ subtracts them, a power of a bracket multiplies them.
3. Work from the inside of the brackets outwards, one law per line.

Simplify an expression (DSE wording: "express your answer with positive indices")

1. Deal with the numbers first.
2. Remove brackets: raise every factor inside to the power.
3. Collect each letter: add indices when multiplying, subtract when dividing.
4. Move any negative index to the other side of the fraction line, so all indices are positive.

Scientific notation

1. Move the decimal point until the number is between 1 and 10.
2. Count the moves: moving left gives a positive power of 10; moving right gives a negative one.
3. When multiplying, multiply the numbers and add the powers; then fix the first part to be between 1 and 10.

Binary to denary (NF)

1. Write the place values from the right: 1, 2, 4, 8, 16, …
2. Add the place values where the digit is 1.
3. Check: the denary number is odd exactly when the last binary digit is 1.

Worked examples

(2a³b⁻²)² = 4a⁶b⁻⁴                square every factor
4a⁶b⁻⁴ ÷ 4a⁻¹b = a^(6−(−1)) b^(−4−1)
               = a⁷ b⁻⁵
               = a⁷ / b⁵          positive indices

(2³)² = 2⁶                  power of a power: multiply
2⁶ × 2⁻⁴ ÷ 2⁰ = 2^(6 − 4 − 0)
              = 2² = 4

numbers: 3 × 4     = 12
powers:  10⁵ × 10⁻² = 10³
12 × 10³ = 1.2 × 10¹ × 10³ = 1.2 × 10⁴     first part must be between 1 and 10

0.000 304   → 3.04       point moves 4 places right → 3.04 × 10⁻⁴
56 000 000  → 5.6        point moves 7 places left  → 5.6 × 10⁷

place values:  16  8  4  2  1
digits:         1  0  1  1  0
16 + 4 + 2 = 22     last digit 0, so even ✓

Common mistakes

• (2a³)² written as 2a⁶; the 2 must be squared too.
• a³ × a⁴ written as a¹² (multiplying indices instead of adding).
• 6 − (−1) worked out as 5.
• Leaving a negative index in the final answer when the question asks for positive indices.
• 12 × 10³ left as the answer; the first part must be less than 10.
• a⁰ = 0 (it is 1).

How the DSE tests it

• Paper 1 A(1) (3 marks): simplify with positive indices. Method marks for each correct law; the answer mark needs positive indices.
• Paper 2: scientific notation, significant figures, and NF binary numbers in Section B.
• Later, the same laws are used for rational indices and logarithms in F4 exponential and logarithmic functions.

Questions you might not ask in class

More Form 3 maths topics