In these lessons, we will learn how to add indices.

**Related Pages**

Exponents

Rules Of Exponents

More Math Lessons

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.

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

Consider:

a^{2} × a^{3} = (a × a) × (a × a × a)

= a^{2 + 3}

= a^{5}

This is the **first law of exponents**:

a^{m} × a^{n} = a^{m + n}

**Example:**

Simplify the following; give your answers in exponent form

a) 3^{3} × 3^{2}

b) x^{5} × x^{3}

**Solution:**

a) 3^{3} × 3^{2} = 3^{3 + 2}

= 3^{5}

b) x^{5} × x^{3} = x^{5 + 3}

= x^{8}

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

a^{m} + a^{n} ≠ 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} × a^{n} = a^{m + n}

The Quotient Rule: a^{m} ÷ a^{n} = a^{m - n}

The Power Rule: (a^{m})^{n} = a^{mn}

**Example:**

Simplify

(3ab^{4})(2a^{2}b)^{3}

(x^{2}y^{4}/2xz)^{3}

**Example of adding exponents**

(2a^{4})(5^{11})

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