Negative indices
We have seen how we can rewrite expressions with negative powers to have positive powers.
For example, if we simplified \dfrac{a^3}{ a^5} using the division law, we would get a^{-2}. Let's expand the example to see why this is the case:
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 \dfrac{a^3}{ a^5} with a positive index as a^{-2}. This gives us the negative index law. When dealing with algebraic bases we follow exact the same approach.
For any base a, \, a^{-x} = \dfrac{1}{a^x}, x \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 1
Find the value of n such that \dfrac{1}{25}=5^n.
Example 2
Simplify the following, giving your answer with a positive index: \dfrac{9x^2}{3x^9}
For any base a,a^{-x} = \frac{1}{a^x}, x\neq 0
That is, when raising a base to a negative power:
Take the reciprocal of the expression
Turn the power into a positive
Fractional bases with negative indices
When raising a fractional base to a negative power we can combine the individual rules we have seen.
To express \left(\dfrac{a}{b}\right)^{-3} with a positive index, we combine the rules for raising fractions to a power with the rule for negative indices: \left(\dfrac{a}{b}\right)^{n}=\dfrac{a^n}{b^n} and a^{-n}=\dfrac{1}{a^n}.
| \displaystyle \left(\dfrac{a}{b}\right)^{-3} | \displaystyle = | \displaystyle \dfrac{a^{-3}}{b^{-3}} | Apply the power to the fraction |
| \displaystyle = | \displaystyle \dfrac{\dfrac{1}{a^{3}}}{\dfrac{1}{b^{3}}} | Apply the negative index law | |
| \displaystyle = | \displaystyle \dfrac{1}{a^{3}}\times\dfrac{b^{3}}{1} | Multiply by the reciprocal | |
| \displaystyle = | \displaystyle \frac{b^{3}}{a^{3}} | Simplify the products | |
| \displaystyle = | \displaystyle \left(\dfrac{b}{a}\right)^{3} | Write as a fraction to a power |
What has happened is we have found the reciprocal of the expression in the question, and turned the power into a positive. Using this trick will save a lot of time.
Examples
Example 3
Simplify the following, giving your answer with a positive index: \left(\dfrac{y}{4}\right)^{-3}
When raising a fraction to a negative power, we flip the fraction and change the power to positive: \left(\dfrac{a}{b}\right)^{-n}=\left(\dfrac{b}{a}\right)^{n}