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Stage 5.1-3

1.02 Non-positive indices

Lesson

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? For example,

\displaystyle x^5\div x^5\displaystyle =\displaystyle x^{5-5}
\displaystyle =\displaystyle 0

To think about what value we can assign to the term x^0, let's write this division problem as the fraction \dfrac{x^5}{x^5}. Since the numerator and denominator are the same, the fraction simplifies to 1. Notice that this will also be the case with \dfrac{k^{20}}{k^{20}} 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, and the result we arrive at by simplifying fractions is 1. This must mean that x^0=1.

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

For any base a, a^0=1. This says that taking the zeroth power of any number will always result in 1.

Examples

Example 1

Evaluate \left(6\times 19\right)^0.

Worked Solution
Create a strategy

Simplify the expression using the rule a^0=1.

Apply the idea

Everything is to the power of 0 so we can apply the rule to the whole expression.

\displaystyle \left(6\times 19\right)^0\displaystyle =\displaystyle 1
Idea summary

For any base a,a^0=1

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

Negative indices

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

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

Expanded form of A to the third power divided by A to the fifth power. Ask your teacher for more information.

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.

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

For any base a, \, a^{-x}=\dfrac{1}{a^x},\,a\neq 0.

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

  • Take the reciprocal of the expression

  • Turn the power into a positive

Examples

Example 2

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

Worked Solution
Create a strategy

Use the negative index law a^{-n}=\dfrac{1}{a^n}.

Apply the idea
\displaystyle 6^{-10}\displaystyle =\displaystyle \frac{1}{6^{10}}Take the reciprocal with a positive power

Example 3

Simplify \dfrac{\left(5^2\right)^9\times 5^6}{5^{40}}, giving your answer in the form a^n.

Worked Solution
Create a strategy

Use index laws to simplify the expression.

Apply the idea
\displaystyle \frac{\left(5^2\right)^9\times 5^6}{5^{40}}\displaystyle =\displaystyle \frac{5^{18}\times 5^6}{5^{40}}Use the power of a power law
\displaystyle =\displaystyle \frac{5^{24}}{5^{40}}Use the multiplication law
\displaystyle =\displaystyle 5^{24-40}Use the division law
\displaystyle =\displaystyle 5^{-16}Evaluate the difference
Idea summary

For any base a,a^{-x}=\dfrac{1}{a^x},a\neq 0That is, when raising a base to a negative power:

  • Take the reciprocal of the expression

  • Turn the power into a positive

Outcomes

MA5.1-5NA

operates with algebraic expressions involving positive-integer and zero indices, and establishes the meaning of negative indices for numerical bases

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