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
Simplify the following expressions.
(a) \( x^5 \times x^3 \) [1]
(b) \( \dfrac{y^{10}}{y^4} \) [1]
(c) \( (z^3)^4 \) [1]
Nova's hint:
Try to Solve it with Nova?
These questions use the index laws (also called laws of exponents).
Ask yourself: 'Which index law applies — am I multiplying, dividing, or raising a power to a power?'
For example, \( a^m \times a^n = a^{m+n} \), and \( (a^m)^n = a^{mn} \).
