Multiplying Exponents Calculator

Apply exponent multiplication rules with step-by-step solutions for same base, same exponent, and power of a power.

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Rule: am × an = am+n

34 × 32

Result

Product of Powers Rule
Result
729
36 = 729

Step-by-Step Solution

am × an = am+n

Exponent Multiplication Rules

When multiplying expressions that involve exponents, there are three fundamental rules that simplify the calculation. These rules are cornerstones of algebra and are used extensively in science, engineering, and computer science.

The Three Key Rules

Product of Powers (Same Base)

When multiplying two powers with the same base, keep the base and add the exponents.

am × an = am+n

Power of a Product (Same Exponent)

When multiplying two powers with the same exponent, multiply the bases and keep the exponent.

an × bn = (ab)n

Power of a Power

When raising an exponential expression to another power, multiply the exponents.

(am)n = amn

Zero Exponent

Any nonzero number raised to the power of zero equals 1.

a0 = 1 (a ≠ 0)

Why Do These Rules Work?

Product of Powers

The rule am × an = am+n follows directly from the definition of exponents. Since am means "a multiplied by itself m times," multiplying by an adds n more factors of a, giving a total of m + n factors.

For example: 23 × 22 = (2 × 2 × 2) × (2 × 2) = 25 = 32.

Power of a Product

The rule an × bn = (ab)n works because you can rearrange the factors. Since an × bn gives you n copies of a and n copies of b, you can pair them as n copies of (a × b).

For example: 32 × 42 = 9 × 16 = 144 = 122 = (3 × 4)2.

Power of a Power

The rule (am)n = amn means raising am to the nth power creates n groups of m factors of a, totaling m × n factors.

For example: (23)2 = (8)2 = 64 = 26 = 23×2.

Common Mistakes to Avoid

  • Adding exponents with different bases: 23 × 32 ≠ 65. You can only add exponents when the bases are the same.
  • Multiplying exponents when adding: am × an = am+n, not amn. You add exponents, not multiply them.
  • Confusing multiplication with addition: am + an ≠ am+n. The addition rule only applies to multiplication of powers.
  • Forgetting the negative exponent: a-n = 1/an, not -an.

Applications

Exponent rules are used in many practical contexts:

  • Scientific Notation: Multiplying numbers in scientific notation uses the same-base rule (powers of 10).
  • Compound Interest: Financial calculations involve exponential growth using power rules.
  • Computer Science: Binary arithmetic and algorithm complexity analysis rely heavily on powers of 2.
  • Physics: Dimensional analysis and unit conversions use exponent manipulation.
  • Engineering: Signal processing and logarithmic scales (decibels) use exponent properties.