Negative indices
A negative index means reciprocal. It does not by itself determine whether the answer is negative or smaller than one. Keep every restriction from the original expression
- Negative power Take the reciprocal. The base must be nonzero and n is a positive whole number \(\displaystyle a^{-n}=\frac1{a^n},\quad a\ne0\)
- Outside coefficient Move only the factor with the negative index \(\displaystyle 5a^{-2}=\frac5{a^2},\quad a\ne0\)
- Negative denominator power Dividing by a reciprocal multiplies by the corresponding positive power. Keep the original restriction \(\displaystyle \frac1{a^{-n}}=a^n,\quad a\ne0\)
- Whole-fraction reciprocal Swap the whole numerator and denominator. Both must be nonzero \(\displaystyle \left(\frac pq\right)^{-1}=\frac qp,\quad p,q\ne0\)
- Combined signed indices Add signed indices in a product and subtract the whole denominator index in a quotient \(\displaystyle a^{-2}\div a^{-5}=a^3,\quad a\ne0\)
Worked example
Simplify using positive indices, for nonzero a and b.
- \(\displaystyle a\ne0,\quad b\ne0\)Retain both original nonzero restrictions
- \(\displaystyle 2\times2=4\)Numerator index on a after squaring
- \(\displaystyle (-1)\times2=-2\)Numerator index on b after squaring
- \(\displaystyle \frac{3a^4b^{-2}}{ab^{-4}}\)Apply the outer power to the numerator
- \(\displaystyle 4-1=3\)Subtract the denominator a index
- \(\displaystyle -2-(-4)=2\)Subtract the denominator b index
- \(\displaystyle 3a^3b^2\)Keep the outside coefficient and original restrictions
Common mistakes
Difficulty levels
Questions about this topic
Can the base be zero?+
No. A negative power requires a reciprocal, and division by zero is undefined
Does a negative index make the answer negative?+
No. A positive base gives a positive value. For a negative base, an odd integer index gives a negative value and an even one gives a positive value
Is every negative power less than one?+
No. A positive base between zero and one has a negative power greater than one. For example, one half to negative two is four
What happens to an outside coefficient?+
It stays where it is unless brackets place it inside the power
Why does a negative denominator power move up?+
Dividing by a reciprocal is multiplying by its corresponding positive power
Why keep a nonzero restriction after simplifying?+
Equivalent forms must refer to the same permitted inputs. A simplified expression can appear defined where the original expression was not
How do I subtract a negative index?+
Subtract the entire signed number. Negative two minus negative five is negative two plus five, which is three
Do I move terms across a plus sign?+
No. The reciprocal rule applies to factors and powers. A sum cannot be rearranged as though it were a product
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 3: Algebra and indices