Multiplying powers with the same base adds the exponents: x³ · x⁴ = x⁷. Counting the factors shows why.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Dividing
Dividing powers with the same base subtracts the exponents: x⁷ ÷ x⁴ = x³.
A power of a power
Raising a power to a power multiplies the exponents: (x³)⁴ = x¹².
The zero exponent
x⁰ = 1 for any non-zero x. It follows from the quotient rule: x³ ÷ x³ is both 1 and x⁰.
Negative exponents
A negative exponent means a reciprocal: x⁻³ = 1 ÷ x³. It continues the same pattern downward.
The rules need the same base
x³ · y⁴ cannot be combined. The rules only apply when the bases match.
The rules count factors
x³·x² is five factors, so x⁵. Dividing cancels factors, so exponents subtract. A power of a power gives repeated groups, so they multiply. Expanding once makes each rule obvious.
The bases must match
x³·y² cannot be combined, because the factors are different things. Applying an exponent law across different bases is the most frequent error, and checking the bases first prevents it.
Zero and negative exponents
x⁰ = 1 follows from x³ ÷ x³ being both x⁰ and 1. And x⁻ⁿ = 1/xⁿ, so a negative exponent means a reciprocal, never a negative value. Both are forced by keeping the laws consistent.
Why they matter here
Exponential functions are unusable without them: rewriting 2^(x+3) as 8·2ˣ needs the addition rule. The exponent laws are what make exponential models computable.
Step 2: Try It Yourself
Tap and try it out.
- Point(0, 1)
Step 3: Watch an Example
One step at a time.
Watch Kofi Simplify a Quotient
Kofi simplifies (x⁵ · x³) ÷ x².
- Step 1
The top multiplies powers of the same base, so the exponents add: x⁸.
Step 4: Your Turn
Practice makes it stick.
The Product
Problem 1 of 2
x³ · x⁴ = xⁿ. What is n?
The Power
Problem 2 of 2
(x³)⁴ = xⁿ. What is n?
Apply the Rules
1 of 8
x² · x⁵ = xⁿ. What is n?
2 of 8
x⁹ ÷ x⁴ = xⁿ. What is n?
3 of 8
(x²)⁶ = xⁿ. What is n?
4 of 8
What is 7⁰?
5 of 8
What is 2⁻³, as a decimal?
6 of 8
x⁴ ÷ x⁶ = xⁿ. What is n?
7 of 8
Match each operation with what it does to exponents.
Tap a card on the left to start.
8 of 8
What is 5⁻², as a decimal?
Step 5: Quick Check
Show what you know.
Question 1 of 2
x⁶ · x³ = xⁿ. What is n?
Question 2 of 2
Why does x⁰ equal 1?
What You Learned
- Multiplying adds exponents; dividing subtracts them.
- A power of a power multiplies them.
- A zero exponent gives 1 and a negative exponent gives a reciprocal.