Indices

The term indices is the plural form of index, which means power or exponent. Thus, we will be dealing with powers (or exponents) of numbers here.

If we have xⁿ for instance, we say the x is the base while the n is the power to which x has been raised. The power means the number of times by which a number is multiplied by itself.

Thus, if we write 5³ it means we are multiplying 5 by itself three times as follows

5³ = 5 * 5 * 5 = 125

x³= x * x * x and

xⁿ = x * x * x * … (up to n times)

Laws of indices

The following laws of indices guide us in the simplification of numbers with powers

Law1              x^m * xⁿ = x^(m+n) (take one base and add up the powers, if the bases are the same)

Law2              x^m ÷ xⁿ = x^(m-n) (take one base and subtract the powers, if the bases are the same)

Law3              ¹/(xⁿ) = x^(-n)

Law4              x⁰ = 1 (anything raised to power zero is 1)

Proof of law4

Law4              x⁰ = 1

Let’s take 3-3 in place of 0

Thus, instead of putting 0 as the power, we insert 3-3 as follows

x^3-3 (from law2 above), this becomes

x³ /x³ = 1 (anything divides itself is 1)

Example1

Simplify 5² * 5³

Solution

5² * 5³ = 5²⁺³ (from law1)

         = 5⁵

Example2

Simplify 2²⁰ ÷ 2⁷

Solution

2²⁰ * 2⁷ = 2^20-7

      = 2¹³

Example3

Simplify x³ * x⁵

Solution

x³ * x⁵ = x³⁺⁵ = x⁸

Example4

Simplify x² ÷ x⁷

Solution

x² ÷ x⁷ = x^2-7

 = x^-5

= ¹/_x⁵