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Multiplying Variables with Exponents — Definition, Formula & Examples

Multiplying variables with exponents means combining terms that share the same base by adding their exponents together. For example, x3⋅x2x^3 \cdot x^2 becomes x5x^5 because you add the exponents 3 and 2.

For any base aa and integer exponents mm and nn, the product am⋅an=am+na^m \cdot a^n = a^{m+n}. This property, known as the product of powers rule, applies only when the bases being multiplied are identical.

Key Formula

am⋅an=am+na^m \cdot a^n = a^{m+n}
Where:
  • aa = The common base (a variable or number)
  • mm = The exponent on the first factor
  • nn = The exponent on the second factor

How It Works

When you multiply two powers with the same base, each factor represents repeated multiplication. Writing out x3⋅x2x^3 \cdot x^2 in expanded form gives (x⋅x⋅x)(x⋅x)(x \cdot x \cdot x)(x \cdot x), which is five xx's multiplied together — that is, x5x^5. Adding the exponents is a shortcut for this counting process. If the bases are different, like x3⋅y2x^3 \cdot y^2, you cannot combine them this way. For expressions with coefficients, multiply the coefficients normally and then apply the exponent rule to the variable parts separately.

Worked Example

Problem: Simplify 4x5⋅3x24x^5 \cdot 3x^2.
Multiply the coefficients: Multiply the numerical parts: 4 times 3.
4×3=124 \times 3 = 12
Add the exponents: Both variable parts have base xx, so add the exponents 5 and 2.
x5⋅x2=x5+2=x7x^5 \cdot x^2 = x^{5+2} = x^7
Combine: Put the coefficient and variable together.
4x5⋅3x2=12x74x^5 \cdot 3x^2 = 12x^7
Answer: 12x712x^7

Why It Matters

This rule is essential every time you simplify polynomial expressions, distribute terms, or solve equations in algebra. You will rely on it constantly in courses from Algebra 1 through Calculus when manipulating formulas and factoring.

Common Mistakes

Mistake: Multiplying the exponents instead of adding them (e.g., writing x3⋅x2=x6x^3 \cdot x^2 = x^6).
Correction: When you multiply same-base powers, you add the exponents: x3⋅x2=x3+2=x5x^3 \cdot x^2 = x^{3+2} = x^5. Multiplying exponents applies to a different rule — raising a power to a power, like (x3)2=x6(x^3)^2 = x^6.