4.3 Indices & Index Laws

Indices (also called powers or exponents) are the official shorthand for repeated multiplication, and ICAS assesses them from Paper E (Year 7) right through to Papers I-J, where fractional indices and index laws appear in some of the hardest questions. Since personal calculators are banned, every index question on ICAS is designed to collapse quickly using the laws — if you are doing long multiplication, you have missed the shortcut.

Index Notation: Base and Power

In 3⁴, the base is 3 and the index (or power or exponent) is 4. It means 3 multiplied by itself 4 times:

3⁴ = 3 × 3 × 3 × 3 = 81

Read it as "three to the power of four". Two cases have special names: 5² is "five squared" and 5³ is "five cubed", from the area of a square and the volume of a cube.

Squares, Cubes and Roots Worth Memorising

ICAS assumes instant recall of the small powers. Learn this table until it is automatic:

Also know the powers of 2 (2, 4, 8, 16, 32, 64, 128, 256, 512, 1024) and of 10 (10, 100, 1000, ...), which feed directly into scientific notation.

The square root undoes squaring: √36 = 6 because 6² = 36. The cube root undoes cubing: ³√125 = 5 because 5³ = 125. ICAS will ask things like "the square root of 144" (12) with the negative-looking distractor included — remember that by convention √ means the positive root.

Index Notation and Prime Factorisation

Any whole number can be written as a product of prime factors, and repeated primes are written in index form:

• 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3²
• 200 = 2³ × 5²
• 1080 = 2³ × 3³ × 5

Use a factor tree and keep dividing by the smallest prime that goes in. ICAS uses these forms to ask questions like "write 72 as a product of prime factors in index notation" or to hide a square root: √144 = √(2⁴ × 3²) = 2² × 3 = 12.

The Index Laws

All the laws below require the same base. Learn them as a set:

Why the zero law works: 3³ ÷ 3³ = 1 (anything divided by itself), but by the division law it is also 3³⁻³ = 3⁰. So 3⁰ must equal 1.

Why negative means reciprocal: 2³ ÷ 2⁵ = 2³⁻⁵ = 2⁻², and directly, (2×2×2)/(2×2×2×2×2) = 1/2². So 2⁻² = 1/4. Crucially, a negative index does not make the answer negative — 2⁻³ = 1/8 is positive. This is the single most common misconception ICAS exploits.

Fractional indices (Papers I-J): a¹ᐟ² = √a and a¹ᐟ³ = ³√a. More generally aᵐᐟⁿ = (ⁿ√a)ᵐ. So 8²ᐟ³ = (³√8)² = 2² = 4. Do the root first — it keeps the numbers small.

Evaluating Expressions with Mixed Laws

Worked example 1. Simplify 4⁵ × 4² ÷ 4⁶.

Add then subtract indices: 4⁵⁺²⁻⁶ = 4¹ = 4.

Worked example 2. Evaluate (2³)² × 2⁻⁴.

(2³)² = 2⁶; then 2⁶ × 2⁻⁴ = 2² = 4.

Worked example 3. Evaluate 5⁰ + 5⁻¹.

5⁰ = 1 and 5⁻¹ = 1/5, so the answer is 1 + 1/5 = 6/5 (or 1.2). ICAS mixes the zero and negative laws like this to check you know the difference.

Worked example 4 (senior). Write 27⁻¹ᐟ³ as a fraction.

27¹ᐟ³ = ³√27 = 3, so 27⁻¹ᐟ³ = 1/3.

Remember the trap from the integers section too: −3² = −9 (square first, then negate) but (−3)² = 9. Brackets change everything.

Scientific Notation

Scientific notation (standard form) writes very large or very small numbers as a × 10ⁿ, where a is a number from 1 up to but not including 10, and n is an integer.

• 6 400 000 = 6.4 × 10⁶ (decimal point moved 6 places left; positive index for big numbers)
• 0.00037 = 3.7 × 10⁻⁴ (decimal point moved 4 places right; negative index for small numbers)

The index counts the moves, and the sign tells you the direction: large numbers get positive indices, small numbers (between 0 and 1) get negative ones.

Worked example. The Sun is about 150 000 000 km from Earth. In scientific notation that is 1.5 × 10⁸ km. A red blood cell is about 0.0000075 m across: 7.5 × 10⁻⁶ m.

To compare two numbers in scientific notation, compare the indices first: 2 × 10⁹ is bigger than 9 × 10⁸ because 10⁹ beats 10⁸. ICAS asks ordering and conversion questions in this form, and the distractors will include versions where the first factor is outside the 1-to-10 range (like 15 × 10⁷, which is not proper scientific notation — it should be 1.5 × 10⁸).

How ICAS Phrases These Questions

• "Simplify, giving your answer in index form" — apply the laws, do not evaluate.
• "Evaluate 2⁻³" — the answer is a positive fraction, 1/8; watch for the −8 distractor.
• "Which is equal to (3²)³?" — 3⁶, not 3⁵ (adding instead of multiplying is the planted error).
• "Express in scientific notation" — check the first factor is between 1 and 10 and the index counts the moves correctly.