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Work with index notation

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Power
\(\displaystyle 5^{3}\)

What does this power mean?

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

Index notation

For a positive integer index, a power represents repeated multiplication. The base is the repeated factor and the index counts its copies. Zero and negative indices are introduced later

\[a^{m}=\underbrace{a\times\cdots\times a}_{m\text{ factors}}\]
  1. Identify the base The base is the number or pronumeral that is repeated as a factor \(\displaystyle 5^{3}\)
  2. Read the index The index shows how many factors of the base are multiplied together \(\displaystyle x^{4}=x\times x\times x\times x\)
  3. Evaluate by repeated multiplication Multiply together as many copies of the base as the positive integer index specifies \(\displaystyle 2^{5}=2\times2\times2\times2\times2=32\)
  4. Track signs and fractions An odd power of a negative base is negative, and a fraction base is raised in the numerator and denominator \(\displaystyle \begin{gathered}(-3)^3=-27\\\left(\frac23\right)^2=\frac49\end{gathered}\)

Worked example

Evaluate the power with a negative base

\[(-2)^{5}\]
  1. \(\displaystyle \begin{gathered}(-2)^5=(-2)\times(-2)\\{}\times(-2)\times(-2)\times(-2)\end{gathered}\)The index 5 means that five factors of -2 are multiplied
  2. \(\displaystyle (-2)\times(-2)=4\)The first pair of negative factors gives a positive 4
  3. \(\displaystyle 4\times(-2)=-8\)Multiplying by the third negative factor makes the partial product negative
  4. \(\displaystyle -8\times(-2)\times(-2)=-32\)Multiplying by the two remaining negative factors gives -32

Common mistakes

Swapping the base and index in three to the fourth power
✕\(\displaystyle \text{base}=4,\quad\text{index}=3\)✓\(\displaystyle \text{base}=3,\quad\text{index}=4\)
For three to the fourth power, the base is 3 and the index is 4: multiply four copies of 3
Evaluating a power as a product
✕\(\displaystyle 2^{4}=2\times4=8\)✓\(\displaystyle 2^{4}=2\times2\times2\times2=16\)
The index counts repeated factors, so 2 to the fourth power has four factors of 2
Dropping the sign of a negative base
✕\(\displaystyle (-3)^{3}=27\)✓\(\displaystyle (-3)^{3}=-27\)
An odd number of negative factors gives a negative product

Difficulty levels

introIdentify a base or index and evaluate a small positive power
standardExpand or rewrite a pronumeral power and evaluate a negative integer base
challengeEvaluate a fraction base or a larger power, sometimes in a repeated growth or decay setting
examChoose the exact value for an odd power of a negative integer from misconception-based options

Questions about this topic

What is the base in a power?+

The base is the repeated factor. In seven cubed, the base is 7 and the index is 3

What does the index count?+

For a positive integer index, it counts how many copies of the base are multiplied. For example, x to the fourth power means four factors of x. Zero and negative indices use extensions of this definition, taught later

Why does the sign depend on the index?+

Negative factors pair to make positive products. An even number of negative factors gives a positive result, while an odd number gives a negative result

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