Search topics

Search

Work with negative indices

Start a set of 8
1 OF 12

A scale factor is 2⁻³. Which value does it represent?

Year 9, Chapter 6, 4 difficulty levels, No calculatorBuild a worksheet

Negative indices

For a nonzero base, a negative integer index means a reciprocal. It does not insert a minus sign into the value. Keep outside coefficients in place. A positive base below 1 gives a value above 1 when raised to a negative index

\[\begin{gathered}a^{-m}=\frac{1}{a^m}\\\frac{1}{a^{-m}}=a^m\quad(a\ne0)\end{gathered}\]
  1. Use the reciprocal This rule requires a nonzero base. Here x ≠ 0 and m is a positive integer \(\displaystyle x^{-3}=\frac{1}{x^{3}}\)
  2. Move only the negative-index factor A coefficient without the negative index stays where it is. Here a ≠ 0. \(\displaystyle 5a^{-2}=\frac{5}{a^{2}}\)
  3. Flip a reciprocal with a negative index The reciprocal of a negative-index power becomes a positive-index power. The original negative power still requires a ≠ 0. \(\displaystyle \frac{1}{a^{-m}}=a^{m}\)
  4. Invert a fraction base A fraction raised to a negative index is inverted before the positive power is evaluated \(\displaystyle \begin{gathered}(\frac23)^{-2}=(\frac32)^2\\=\frac94\end{gathered}\)

Worked example

Rewrite using positive indices. Assume p ≠ 0.

\[6p^{-3}q^{2}\]
  1. \(\displaystyle p^{-3}=\frac{1}{p^{3}}\)The negative index changes p to its reciprocal with index 3.
  2. \(\displaystyle \begin{gathered}6p^{-3}q^2\\=6\times\frac{1}{p^3}\times q^2\end{gathered}\)The coefficient 6 stays in the numerator and q has a positive index already
  3. \(\displaystyle \begin{gathered}6\times\frac{1}{p^3}\times q^2\\=\frac{6q^2}{p^3}\end{gathered}\)Write the factors with the coefficient and positive indices in their new positions

Common mistakes

Keeping the negative index as a negative value
✕\(\displaystyle x^{-4}=-x^{4}\)✓\(\displaystyle x^{-4}=\frac{1}{x^{4}}\)
A negative index means a reciprocal, not a negative value. Assume x ≠ 0.
Moving a base without changing its index
✕\(\displaystyle x^{-3}=\frac{1}{x^{-3}}\)✓\(\displaystyle x^{-3}=\frac{1}{x^{3}}\)
When a factor crosses the fraction bar, its index changes sign and becomes positive here. Assume x ≠ 0.
Applying the negative index to a coefficient
✕\(\displaystyle 3x^{-2}=(3x)^{-2}=\frac{1}{9x^{2}}\)✓\(\displaystyle 3x^{-2}=\frac{3}{x^{2}}\)
Only x carries the negative index, so the coefficient 3 stays unchanged. Assume x ≠ 0.

Difficulty levels

introRewrite a single negative index as a reciprocal with a positive index
standardRewrite a reciprocal form with a coefficient and multiple pronumeral factors
challengeEvaluate small fraction bases above or below 1 raised to a negative index exactly
examChoose the positive-index form from options containing common negative-index errors

Questions about this topic

What does a negative index mean?+

For a nonzero base, it means the reciprocal of the same base raised to the corresponding positive index. For example, 2⁻³=1/8.

Does a negative index make a value negative?+

No. It changes a power into a reciprocal. A positive base therefore still has a positive value

What happens to a coefficient next to a negative-index factor?+

The coefficient stays unchanged unless it also carries the index. In 3x⁻², the 3 stays in the numerator; assume x ≠ 0.

Does a negative index always give a value below 1?+

No. For example, (2/3)⁻²=9/4, which is above 1. The reciprocal rule determines the value, not the sign of the exponent alone

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 9 Mathematics, chapter 6: Indices and surds

In this chapter

Work with index notationExpress a number in prime-factor index formSimplify using index laws 1 and 2Simplify using index law 3 and the zero powerExpand using index laws 4 and 5Work with negative indicesSolve an exponential equation by matching basesCombine positive and negative integer indicesScientific notationMultiply and divide in scientific notationSI prefixes and scientific timescalesRound to decimal places and significant figuresRecord measurements and compare instrumentsRelate fractional indices to roots (optional extension)Locate exact square roots on the real number lineSimplify, compare and operate with surds (optional practice)