## What are Indices?

In Mathematics, Indices are simply “repeated multiplication”. There are 6 Index laws which needs to be remembered. A basic example showing each part of the equation is shown below:

## The Index Laws

**First Index Law:**

a^{m} × a^{n} = a^{m + n}

Example:

a^{5} × a^{3} = a^{8}

**Second Index Law:**

a^{m} / a^{n} = a^{m – n}

Example:

a^{5} / a^{3} = a^{2}

**Third Index Law:**

a^{0} = 1 (where a ≠ 0)

Example:

Step 1: Simplify the following expression:

4^{3} / 4^{3}

Step 2: Use the second law of indices:

4^{3} / 4^{3} = 4 ^{3-3} = 4^{0}

Step 3: Expand the numerator and denominator:

4^{3}/4^{3} = (4 × 4 × 4) / (4 × 4 × 4)

Step 4: We cancel the “4” and are left with the following:

(1 × 1 × 1) / (1 × 1 × 1) = 1

Step 5: Therefore 4^{0} = 1

**Fourth Index Law:**

(a^{m}) = a^{m × n}

Example:

(a^{5})^{3} = a^{15}

**Fifth Index Law:**

(a × b)^{m} = a^{m} × b^{m}

Example:

(5 × 3)^{5} = 5^{5} × 3^{5}

**Sixth Index Law:**

(a / b)^{m} = a^{m} / b^{m}

Example:

(5 / 6)^{3} = (5/6) × (5/6) × (5/6)

= (5 × 5 × 5) / (6 × 6 × 6)

= 5^{3} / 6^{3}

**Negative Indices:**

a^{-n} = 1 / a^{n} (where a≠0)

Example:

1 / 6^{3} = 6^{0}/6^{3}

= 6^{0-3}

= 6^{-3}

## Roots

Roots are the opposite of exponents that we have been looking at above. You can reverse an exponent with a “root”. For example, if you square 3 you will get 9. If you then square root 9 you will get 3.

√a = a^{1/2}

√9 = 3

^{3}√a = a ^{1/3}

^{3}√64 = 4

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