But what happens in a case where the power in the denominator is greater than the power in the numerator? For example, if we simplify $$x8x10 using the division law, we get $$x8−10=x−2. We are left with a negative power. But what does this mean? Let's expand this example to find out.

Worked Example

Simplify the following using index laws, expressing your answer with a positive index: $$x8÷x10.

Think: Let's write this in expanded form.

Using the division law, we would get $$x8−10=x−2. We can see in the picture above that when we cancel out the common factors, the numerator is $$1 and the denominator is $$x2. So the negative power of $$x−2 can be expressed with a positive index as $$1x2.

Do: $$x8x10=1x2

Reflect: We can see that, in general, $$a−n is the reciprocal (that is, the flipped version) of $$an. So $$a−n actually represents a fraction, not a negative number.

The negative index law

$$a−n=1an, where $$a≠0.

Common Fractions

Consider the number represented by $$2−3. What is this number?

Well we can rewrite it in positive index form by taking the reciprocal: $$2−3=123.

But $$123=18. So we have found another way to represent the fraction $$18, which is $$2−3=18.

Examples

Question 1

Express $$3−1 as a fraction in simplest form.

Question 2

Express $$2−8 in the form $$1xy, using positive indices.

Question 3

Express $$164 with a negative index.