2³ × 2⁴ is three 2s times four 2s, which is seven 2s. So the exponents add.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Dividing subtracts
2⁵ ÷ 2² cancels two of the 2s, leaving three. So the exponents subtract.
Why anything to the power 0 is 1
2³ ÷ 2³ is 1, and the rule gives 2⁰. So 2⁰ must be 1.
Negative exponents
2⁻² is 1 ÷ 2², which is 1/4. Carrying the pattern below zero forces it.
The laws are just counting factors
x³ × x² is (x·x·x) × (x·x), five factors in all, so x⁵. The rule "add the exponents" is a shortcut for counting. Writing the factors out once makes every exponent law obvious rather than arbitrary.
Dividing subtracts
x⁵ ÷ x² has five factors on top and two below; two cancel, leaving three. So x⁵ ÷ x² = x³. Subtracting exponents is cancellation, which is why the rule only works when the bases are the same.
A power of a power multiplies
(x²)³ means x² × x² × x², which is six factors: x⁶. The exponents multiply because you have three groups of two. Again, expanding once shows why, and the shortcut follows.
The laws need the same base
x³ × y² cannot be simplified — the factors are different things. All the exponent laws require a common base, and applying them across different bases is the most common error in the topic.
Step 2: Try It Yourself
Tap and try it out.
4 × 4 = 16
4 rows of 4 makes 16.
Step 3: Watch an Example
One step at a time.
Watch Ana Simplify 3² × 3⁴
Ana multiplies two powers of the same base.
- Step 1
She counts the factors: two 3s times four 3s.
Step 4: Your Turn
Practice makes it stick.
The Product
Problem 1 of 2
2³ × 2² = 2 to what power?
The Quotient
Problem 2 of 2
5⁷ ÷ 5⁴ = 5 to what power?
Count the Factors
1 of 4
4² × 4³ = 4 to what power?
2 of 4
6⁸ ÷ 6³ = 6 to what power?
3 of 4
What is 7⁰?
4 of 4
What is 2⁻² as a fraction with top 1? Give the bottom number.
Step 5: Quick Check
Show what you know.
Question 1 of 2
3⁵ × 3² = 3 to what power?
Question 2 of 2
Why does multiplying powers add the exponents?
What You Learned
- Multiplying powers of the same base adds the exponents.
- Dividing subtracts them.
- Anything to the power 0 is 1, and a negative exponent is a reciprocal.