🔢 Raising a Monomial to a Power
Learn the powerful rules that make working with exponents easy!
What is Raising to a Power?
When we raise a monomial to a power (возведение в степень), we multiply that monomial by itself a certain number of times. For example, (2x)³ means 2x · 2x · 2x.
In this lesson, we'll discover three important rules that make this process much easier!
📐 Rule 1: Power of a Product
(ab)n = an · bn
Rule Statement: To raise a product to a power, raise each factor to that power and multiply the results.
Proof:
Let's prove this with (ab)⁴:
(ab)⁴ = (ab)(ab)(ab)(ab)
= (a · a · a · a)(b · b · b · b)
= a⁴ · b⁴
Each factor gets raised to the power!
Extension to More Factors:
- (abc)n = anbncn
- (abcd)n = anbncndn
The rule applies to any number of factors!
Visual: Power of a Product
\( (ab)^4 \)
Write out the repeated multiplication
\( = (ab)(ab)(ab)(ab) \)
Regroup by factor
\( = (a \cdot a \cdot a \cdot a)(b \cdot b \cdot b \cdot b) \)
Apply exponent notation
\( = a^4 \cdot b^4 \)
Each factor inside the parentheses is raised to the power separately.