Adding Indices

Rules of Indices
The rules of indices (also known as the rules of exponents) are fundamental principles that govern how to manipulate powers and roots in mathematics. They are essential for simplifying expressions, solving equations, and working with scientific notation.

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.

 

Indices/Exponents Worksheets
Practice your skills with the following worksheets:
Printable & Online Indices/Exponents Worksheets

Free Exponent Games Online
Multiply Exponents	Divide Exponents	Power of a Power
Power of a Product	Negative Exponents	Fractional Exponents

 

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}

Here are the primary rules of indices:
Let a and b be real numbers, and m and n be integers (unless specified otherwise).

1. Multiplication Rule (Product Rule):
   When multiplying powers with the same base, add the exponents.
   a^m × aⁿ = a^m+n
   
2. Division Rule (Quotient Rule):
   When dividing powers with the same base, subtract the exponents.
   a^m ÷ aⁿ = a^m-n (where a ≠ 0)
   
3. Power of a Power Rule:
   When raising a power to another power, multiply the exponents.
   (a^m)ⁿ = a^mn
   
4. Power of a Product Rule:
   When raising a product to a power, raise each factor to that power.
   (ab)ⁿ = aⁿbⁿ
   
5. Power of a Quotient Rule:
   When raising a quotient to a power, raise both the numerator and the denominator to that power.
   \(\left( \frac{a}{b} \right)^n=\frac{a^n}{b^n}\) (where b ≠ 0)
   
6. Zero Exponent Rule:
   Any non-zero base raised to the power of zero is equal to 1.
   a⁰ =1 (where a ≠ 0)
   
7. Negative Exponent Rule:
   A base raised to a negative exponent is equal to its reciprocal raised to the positive exponent.
   \(a^{-n}=\frac{1}{a^n}\) (where a ≠ 0)
   
8. Fractional Exponent Rule:
   A fractional exponent indicates a root.
   \(a^{\frac{1}{n}}=\sqrt[n]{a}\)
   
9. General Fractional Exponent Rule:
   A fractional exponent where the numerator is not 1.
   \(a^\frac{m}{n}=\left( \sqrt[n]{a} \right)^m =\sqrt[n]{a^m}\)
   

Index rules - add and subtract indices
Basic look at the first two index laws.

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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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