Negative Indices
A negative index means a reciprocal: \(4^{-2} = \frac{1}{4^2} = \frac{1}{16}\). This NSW Year 9 Core maths subtopic builds the idea from the dividing-by-3 pattern \(3^2, 3^1, 3^0, \ldots\) and from the division law, under NESA outcome MA5‑IND‑C‑01, then uses it with pronumerals.
You will turn negative indices into reciprocals and write answers with positive indices, noticing that in \(7c^{-2} = \frac{7}{c^2}\) the coefficient is not affected, while in \((7c)^{-1} = \frac{1}{7c}\) it is. You will raise fractions to a negative power by flipping them, as in \(\left(\frac{h}{5}\right)^{-2} = \frac{25}{h^2}\), combine negative indices with a power of a power, and divide by a term with a negative index. Negative indices are how scientists write very small quantities. It is part of Year 9 Core Indices and Scientific Notation.
Theory
A negative index means a reciprocal: \(a^{-n} = \dfrac{1}{a^n}\). This Year 9 maths page shows where the rule comes from, how to evaluate negative powers, how to write answers with positive indices, and why a fraction to a negative power turns upside down.
A negative index tells you to take the reciprocal. The reciprocal of a number is \(1\) divided by it, so the reciprocal of \(5\) is \(\dfrac{1}{5}\).
So \(a^{-n} = \dfrac{1}{a^n}\). For example \(3^{-2} = \dfrac{1}{3^2} = \dfrac{1}{9}\). The answer is a fraction, not a negative number.
The index only moves what it touches. In \(5m^{-2}\) only the \(m\) goes under the fraction line: \(\dfrac{5}{m^2}\). In \((5m)^{-2}\) the whole bracket goes: \(\dfrac{1}{25m^2}\).
A fraction to a negative power turns upside down and the index becomes positive: \(\left(\dfrac{2}{7}\right)^{-1} = \dfrac{7}{2}\).
"Write with positive indices" means rewrite the answer so that no index is negative.
For any base \(a\) except \(0\):
Why. By the division law, \(\dfrac{a^2}{a^5} = a^{2-5} = a^{-3}\). Cancelling gives \(\dfrac{1}{a^3}\). Both are the same thing.
A fraction to a negative power:
| Expression | What moves | Answer |
|---|---|---|
| \(y^{-4}\) | the \(y^4\) goes underneath | \(\dfrac{1}{y^4}\) |
| \(6y^{-4}\) | only the \(y^4\) moves | \(\dfrac{6}{y^4}\) |
| \(6xy^{-4}\) | only the \(y^4\) moves | \(\dfrac{6x}{y^4}\) |
| \((6y)^{-4}\) | the whole bracket moves | \(\dfrac{1}{(6y)^4} = \dfrac{1}{1296y^4}\) |
How to write an answer with positive indices
- Look for brackets. Decide what each negative index touches.
- Move that part to the other side of the fraction line and make its index positive. A fraction in a bracket just flips.
- Raise every factor in any bracket to its power.
- Tidy up: numbers that did not move stay where they are. Check no index is negative.
(i) \(5^{-2}\)
(ii) \(\left(\dfrac{3}{4}\right)^{-2}\)
(i) Write the reciprocal to make the index positive, then work out the power.
| \(5^{-2}\) | \(=\) | \(\dfrac{1}{5^2}\) |
| \(\) | \(=\) | \(\mathbf{\dfrac{1}{25}}\) |
(ii) A negative index on a fraction turns it upside down.
| \(\left(\dfrac{3}{4}\right)^{-2}\) | \(=\) | \(\left(\dfrac{4}{3}\right)^{2}\) |
| \(\) | \(=\) | \(\dfrac{4^2}{3^2}\) |
| \(\) | \(=\) | \(\mathbf{\dfrac{16}{9}}\) |
Both answers are positive. The minus in the index does not make the answer negative.
(i) \(3x^{-4}\)
(ii) \((2y)^{-3}\)
(iii) \((4k^3)^{-2}\)
(i) No bracket, so only the \(x\) moves. The \(3\) stays on top.
| \(3x^{-4}\) | \(=\) | \(3 \times \dfrac{1}{x^4}\) |
| \(\) | \(=\) | \(\boldsymbol{\dfrac{3}{x^4}}\) |
(ii) The bracket moves underneath as a whole. Then expand it.
| \((2y)^{-3}\) | \(=\) | \(\dfrac{1}{(2y)^3}\) |
| \(\) | \(=\) | \(\boldsymbol{\dfrac{1}{8y^3}}\) |
(iii) Move the bracket, then raise every factor. Multiply the indices for \(k^3\).
| \((4k^3)^{-2}\) | \(=\) | \(\dfrac{1}{(4k^3)^2}\) |
| \(\) | \(=\) | \(\boldsymbol{\dfrac{1}{16k^6}}\) |
(i) \(\left(\dfrac{a}{4}\right)^{-2}\)
(ii) \(\left(\dfrac{-2}{p}\right)^{-3}\)
(i) Flip the fraction and make the index positive. Then raise the top and the bottom.
| \(\left(\dfrac{a}{4}\right)^{-2}\) | \(=\) | \(\left(\dfrac{4}{a}\right)^{2}\) |
| \(\) | \(=\) | \(\boldsymbol{\dfrac{16}{a^2}}\) |
(ii) Flip it the same way. The index \(3\) is odd, so the minus sign stays.
| \(\left(\dfrac{-2}{p}\right)^{-3}\) | \(=\) | \(\left(\dfrac{p}{-2}\right)^{3}\) |
| \(\) | \(=\) | \(\dfrac{p^3}{(-2)^3}\) |
| \(\) | \(=\) | \(\dfrac{p^3}{-8}\) |
| \(\) | \(=\) | \(\boldsymbol{-\dfrac{p^3}{8}}\) |
Divide the numbers. Then subtract the indices, even though one is negative.
| \(8 \div 2\) | \(=\) | \(4\) |
| \(y^4 \div y^{-3}\) | \(=\) | \(y^{4-(-3)}\) |
| \(\) | \(=\) | \(y^{4+3} = y^7\) |
Subtracting a negative is the same as adding.
| \(\dfrac{8y^4}{2y^{-3}}\) | \(=\) | \(\boldsymbol{4y^7}\) |
Another way to see it: \(\dfrac{1}{y^{-3}} = y^3\), so the \(y^3\) moves up to the top and joins the \(y^4\).
Common pitfalls
Frequently asked questions
What does a negative index mean?
A negative index means take the reciprocal: \(a^{-n} = \dfrac{1}{a^n}\), so \(3^{-2} = \dfrac{1}{9}\).
Does a negative power make a number negative?
No. A negative power gives a fraction, not a negative number. \(5^{-2} = \dfrac{1}{25}\), which is positive.
How do you get rid of a negative index?
Move the part the index touches to the other side of the fraction line and make the index positive: \(3x^{-4} = \dfrac{3}{x^4}\).
What happens to a fraction with a negative power?
It turns upside down and the index becomes positive: \(\left(\dfrac{3}{4}\right)^{-2} = \left(\dfrac{4}{3}\right)^2 = \dfrac{16}{9}\).
Why is a to the power negative n equal to 1 over a to the n?
\(\dfrac{a^2}{a^5} = a^{-3}\) by the division law, and cancelling gives \(\dfrac{1}{a^3}\). Both are the same, so \(a^{-3} = \dfrac{1}{a^3}\).