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Middle Years

5.04 Non-positive indices


The zero index

What happens if we want to divide one term by another and when we perform the subtraction and we are left with a power of $0$0? For example,

$x^5\div x^5$x5÷​x5 $=$= $x^{5-5}$x55
  $=$= $x^0$x0

To think about what value we can assign to the term $x^0$x0, let's write this division problem as the fraction $\frac{x^5}{x^5}$x5x5. Since the numerator and denominator are the same, the fraction simplifies to $1$1. Notice that this will also be the case with $\frac{k^{20}}{k^{20}}$k20k20 or any expression where we are dividing like bases whose powers are the same.

So the result we arrive at by using index laws is $x^0$x0, and the result we arrive at by simplifying fractions is $1$1. This must mean that $x^0=1$x0=1.

There is nothing special about $x$x, so we can extend this observation to any base. This result is summarised by the zero power law.


The zero power law

For any base $a$a,


This says that taking the zeroth power of any number will always result in $1$1.


Negative indices

So far we have looked at expressions of the form $\frac{a^m}{a^n}$aman where $m>n$m>n and where $m=n$m=n, and how to simplify them using the division rule and also the zero power rule.


But what happens when $m$m is smaller than $n$n? For example, if we simplified $a^3\div a^5$a3÷​a5 using the division law, we would get $a^{-2}$a2. So what does a negative index mean? Let's expand the example to find out:

Remember that when we are simplifying fractions, we are looking to cancel out common factors in the numerator and denominator. Remember that any number divided by itself is $1$1.

So using the second approach, we can also express $a^3\div a^5$a3÷​a5 with a positive index as $\frac{1}{a^2}$1a2. The result is summarised by the negative index law.


Negative index law

For any base $a$a,


That is, when raising a base to a negative power:

  • Take the reciprocal of the expression
  • Turn the power into a negative


Practice questions

Question 1

Evaluate $\left(6\times19\right)^0$(6×19)0.

Question 2

Express $6^{-10}$610 with a positive index.

Question 3

Simplify $\frac{\left(5^2\right)^9\times5^6}{5^{40}}$(52)9×56540, giving your answer in the form $a^n$an.

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