Multiplication and Division of Powers
Learn the fundamental rules for working with exponents when multiplying and dividing powers with the same base.
What Are Powers?
A power is a shorthand way to write repeated multiplication of the same number:
an = a × a × a × ... × a
(n factors of a)
Where:
- a is called the base (основание)
- n is called the exponent or power (показатель степени)
📐 Multiplying Powers with the Same Base
Derivation from Definition
Let's understand why the rule works by expanding the powers:
am × an = ?
= (a × a × ... × a) × (a × a × ... × a)
↑ m factors ↑ n factors
= a × a × a × ... × a
↑ (m + n) factors total
Rule for Multiplication of Powers (Умножение степеней)
am × an = am+n
To multiply powers with the same base, keep the base and ADD the exponents.
2³ × 2⁴ = ?
= (2 × 2 × 2) × (2 × 2 × 2 × 2)
Interactive visualization — available in the full lesson
3 factors
×
Interactive visualization — available in the full lesson
4 factors
= 2 × 2 × 2 × 2 × 2 × 2 × 2
Interactive visualization — available in the full lesson
← 7 factors total
= 23+4 = 27
= 128
Extending to Three or More Powers
The rule extends to any number of factors:
am × an × ak = am+n+k
Example: 2² × 2³ × 2⁴ = 22+3+4 = 2⁹ = 512