Product Rule for Exponents

Product Rule for Exponents
The Product Rule for Exponents states that when we multiply two powers that have the same base we can add the exponents. The base remains the same.

Formula:
a^m × aⁿ = a^{(m+n)
Where:
(a) is the base (any real number or variable).
(m) and (n) are the exponents (any real numbers).}

How to Apply the Product Rule:

1. Check if the bases are the same: The product rule only applies when the numbers or variables being multiplied have the same base.
   
2. Add the exponents: If the bases are the same, keep the base and add the exponents together.
   

The following diagram shows the product rule for exponents. Scroll down the page for more examples and solutions for the exponential rules.

 

Rules of Exponents
Refer to the following for more Rules of Exponents:
Rules of Exponents

• Worksheets to Practice Exponent Rules

  Printable
  Product Rule
  Quotient Rule
  Power Rule
  Negative Exponents
  Fractional exponents
  Laws of Exponents
  

  Online
  Apply Law of Exponents (multiply)
  Apply Law of Exponents (mixed)
  

Examples:
Numerical Bases:
(2³ × 2² = 2⁽³⁺²⁾ = 2⁵ = 32)
(2 × 2 × 2) × (2 × 2) = 8 × 4 = 32

(5 × 5⁴ = 5⁽¹⁺⁴⁾ = 5⁵ = 3125)
(5) × (5 × 5 × 5 × 5) = 5 × 625 = 3125

((-3)² × (-3)³ = (-3)⁽²⁺³⁾ = (-3)⁵ = -243)
((-3) × (-3)) × ((-3) × (-3) × (-3)) = (9) × (-27) = -243

Variable Bases:
x⁴ × x⁵ = x⁴⁺⁵ = x⁹
y × y⁷ = y¹⁺⁷ = y⁴
a² × a^-3 = a^2+(-3) = a^-1 = 1/a
b^1/2 × b^3/2 = b^1/2+3/2 = b^4/2 = b^1/2

With Coefficients:
When terms with coefficients and the same base are multiplied:

1. Multiply the coefficients.
2. Apply the product rule to the variables (add the exponents).
   (3x² × 4x³ = (3 × 4)(x²⁺³) = 12x⁵)
   (-2y⁵ × 7y^-2 = (-2 × 7)(y^5+(-2)) = -14y³)
   

How to use the product rule for exponents to simplify an expression?
To use the product rule for exponents to simplify an expression, you need to identify terms that have the same base and are being multiplied.

Steps:

1. Identify Terms with the Same Base: Look for factors in the expression that have the same base (either a number or a variable).
2. Apply the Product Rule: For each set of terms with the same base that are being multiplied, add their exponents.
3. Simplify Coefficients (if any): If the terms have numerical coefficients, multiply those coefficients together.
4. Combine Simplified Terms: Write the simplified expression with the multiplied coefficients and the variables with their new (summed) exponents.

Product Rule for Exponents
This video develops the Product Rule for Exponents.

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Algebra - Product rule for exponents
This video show how the product rule for exponents can be used to simplify an expression. Remember that for the product rule we add the exponents.

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Product rule for exponents
This video demonstrates how to use the product rule for exponents.

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Exponent Rules, Multiplying.

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Exponents - Product Rule.

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Simplify Exponential Expressions - Product and Power Rule
This video provides several examples of how to simplify exponential expressions. The product and power rule of exponents are emphasized.

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