Index Law Basics
Grade 4What is an index?
An index (or power or exponent) is the small raised number that tells you how many times to multiply a base by itself. In a³, the base is a and the index is 3, so it means a × a × a.
Most students are fine with this until expressions start combining. That is where the index laws save you from writing everything out the long way.
The three laws you need
a^m × a^n = a^(m+n)
a^m ÷ a^n = a^(m−n)
(a^m)^n = a^(m×n)
These laws only work when the bases are the same. You cannot apply them to something like a³ × b², because a and b are different bases.
Worked example — multiply and divide
EXAMPLE
Simplify: (a) x⁵ × x³ (b) y⁸ ÷ y² (c) (p³)⁴
(a) Same base, multiplying — add the powers: x⁵ × x³ = x⁸.
(b) Same base, dividing — subtract the powers: y⁸ ÷ y² = y⁶.
(c) Power raised to a power — multiply the powers: (p³)⁴ = p¹².
Watch out — common slips
x² × x³ is not x⁶. You add the powers (giving x⁵), you do not multiply them. Multiplying powers is what you do with (x²)³.
Try this one
Work out the value of each expression when \( x = 3 \).
(a) \( x^2 + 7 \) [1]
(b) \( 2x^3 - 10 \) [2]
Nova's hint:
Try to Solve it with Nova?
This is about substitution into expressions with indices.
Ask yourself: 'Do I need to apply the power to \( x \) before multiplying by the coefficient?'
For example, if \( x = 2 \), then \( 3x^2 = 3 \times 2^2 = 3 \times 4 = 12 \). Always deal with the power before multiplying.
