Multiplying Powers – Adding Exponents with the Same Base

Note: this page is about multiplying two powers that share the same base (like x³ × x²) – a different skill from multiplying a plain number by a power of 10, which is covered on its own dedicated page in the Arithmetic Operations section.

When two powers share the same base, multiplying them together is remarkably simple: just add the exponents. This is called the Product Law: x^a × x^b = x^a+b. It works because x^a means “a copies of x multiplied together,” and x^b means “b more copies” – multiplying them just combines all the copies into one long chain of (a+b) x's.

The Product Law

x^a × x^b = x^a+b, as long as both powers share the same base x.

Simplify 5² × 5³ as a single power of 5.

Add the exponents: 2 + 3 = 5. So 5² × 5³ = 5⁵.

Checking with Real Numbers

Compute 2² × 2³ directly, then check it matches 2⁵.

2² × 2³ = 4 × 8 = 32. And 2⁵ = 32. They match! 32.

Real-Life Application

Algebra: combining variables raised to powers in an expression.
Scientific notation: multiplying two measurements both written with powers of 10.
Computing: combining storage or processing capacities expressed as powers of 2.

Key Takeaways

To multiply powers with the same base, add the exponents: x^a × x^b = x^a+b.
The bases must match for this rule to apply directly.
Dividing powers, covered next, works the same way but with subtraction.

Practice: Multiplying Powers

Multiplying Powers