Adding Indices

The following diagrams show the rules of indices or laws of indices. Scroll down the page for more examples and solutions on how to use the rules of indices.

• Rule of Indices Worksheets

  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)

When multiplying numbers in exponent notation with the same base, we can add the exponents.

Consider:
a² × a³ = (a × a) × (a × a × a)
= a^{2 + 3}
= a⁵

This is the first law of exponents:
a^m × aⁿ = a^{m + n}

Example:
Simplify the following; give your answers in exponent form
a) 3³ × 3²
b) x⁵ × x³

Solution:
a) 3³ × 3² = 3^{3 + 2}
= 3⁵
b) x⁵ × x³ = x^{5 + 3}
= x⁸

Common Error: The first law of exponents does NOT apply to addition of numbers in exponent notation.

a^m + aⁿ ≠ a^{m + n}

Scroll down the page for more examples and solutions of the first law of exponents and also the other laws of exponents.

Index rules - add and subtract indices

Basic look at the first two index laws.

Properties of Exponents

The Product Rule: a^m × aⁿ = a^{m + n}
The Quotient Rule: a^m ÷ aⁿ = a^{m - n}
The Power Rule: (a^m)ⁿ = a^mn

Example:
Simplify
(3ab⁴)(2a²b)³
(x²y⁴/2xz)³

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Example of adding exponents

(2a⁴)(5¹¹)

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