Multiplying like bases adds the exponents: x³ · x⁴ = x⁷, because that is seven x factors.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Dividing
Dividing like bases subtracts them: x⁷ ÷ x⁴ = x³. Four factors cancel.
A power of a power
(x³)⁴ multiplies them: four copies of three factors is twelve factors.
Why they are not arbitrary
Every rule is just counting factors. Writing them out once makes them impossible to mix up.
The laws come from counting factors
x³·x² is five factors of x, so x⁵. Dividing cancels factors, so exponents subtract. A power of a power gives repeated groups, so exponents multiply. Expanding once makes every rule self-evident.
Same base required
The laws only apply when the bases match. x³·y² cannot be simplified. Applying an exponent rule across different bases is the most frequent error, and checking the bases first prevents it entirely.
Zero and negative exponents
x⁰ = 1 because x³ ÷ x³ is both x⁰ and 1. And x⁻ⁿ = 1/xⁿ, so a negative exponent is a reciprocal, not a negative value. Both definitions are forced by requiring the laws to keep working.
Fractional exponents are roots
x^(1/2) = √x, because squaring it gives x¹. In general x^(m/n) is the nth root of x to the mth power. The notation unifies roots and powers into a single system with one set of rules.
Step 2: Try It Yourself
Tap and try it out.
Before: 2
After: 8
The factor is more than 1, so the bar got longer.
Step 3: Watch an Example
One step at a time.
Watch Dev Count Factors
Dev checks that x² · x³ = x⁵ by writing it out.
- Step 1
x² is x times x.
Step 4: Your Turn
Practice makes it stick.
Multiplying
Problem 1 of 2
x⁴ · x⁵. What is the exponent?
A Power of a Power
Problem 2 of 2
(x²)⁵. What is the exponent?
Which Operation on the Powers
1 of 8
x³ · x⁶. Exponent?
2 of 8
x⁹ ÷ x⁴. Exponent?
3 of 8
(x⁴)³. Exponent?
4 of 8
x⁵ ÷ x⁵. Exponent?
5 of 8
2³. What is the value?
6 of 8
Match each expression to what happens to the exponents.
Tap a card on the left to start.
7 of 8
(x³)³. Exponent?
8 of 8
x⁸ ÷ x². Exponent?
Step 5: Quick Check
Show what you know.
Question 1 of 1
x⁷ · x². Exponent?
What You Learned
- Multiplying like bases adds the exponents; dividing subtracts them.
- A power of a power multiplies them.
- Every rule is just counting factors, which is why writing them out settles any doubt.