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Express a number in prime-factor index form

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Whole number
\(\displaystyle 36\)

Which is a complete prime factorisation in index form?

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

Prime factors in index form

Prime factorisation writes a whole number greater than 1 as a product of primes. In the formula, each p is a distinct prime and its positive integer index a counts its occurrences

\[N=p_{1}^{a_{1}}\times p_{2}^{a_{2}}\times\cdots\]
  1. Repeated prime division In this method, divide by a prime factor at each step. A factor tree may instead split into composite factors, provided they are split further \(\displaystyle 84\div2=42\)
  2. Divide until one Continue dividing by prime factors until the quotient is 1 \(\displaystyle \begin{gathered}84\div2=42\\42\div2=21\\21\div3=7\\7\div7=1\end{gathered}\)
  3. Count repeated primes The number of occurrences of a prime becomes its index \(\displaystyle 2\times2\times3=2^{2}\times3\)
  4. Keep every prime The product of the prime powers must equal the original whole number \(\displaystyle 60=2^{2}\times3\times5\)

Worked example

Express 252 as a product of prime factors in index form

\[252\]
  1. \(\displaystyle \begin{gathered}252\div2=126\\126\div2=63\end{gathered}\)Divide by 2 twice, recording two factors of 2
  2. \(\displaystyle \begin{gathered}63\div3=21\\21\div3=7\end{gathered}\)Divide by 3 twice, recording two factors of 3
  3. \(\displaystyle 252=2\times2\times3\times3\times7\)The remaining quotient 7 is prime, so the complete prime product is shown
  4. \(\displaystyle \begin{gathered}2\times2\times3\times3\times7\\=2^2\times3^2\times7\end{gathered}\)Group equal prime factors and use their counts as indices

Common mistakes

Leaving a composite factor
✕\(\displaystyle 84=2^{2}\times21\)✓\(\displaystyle 84=2^{2}\times3\times7\)
Every factor in a prime factorisation must be prime, so a composite factor must be split further
Miscounting a prime index
✕\(\displaystyle 72=2^{2}\times3^{2}\)✓\(\displaystyle 72=2^{3}\times3^{2}\)
The index of a prime is the number of times that prime occurs in the factorisation
Omitting a prime factor
✕\(\displaystyle 90=2\times3^{2}\)✓\(\displaystyle 90=2\times3^{2}\times5\)
Every prime factor must be included so that the product equals the original number

Difficulty levels

introFactor a small number with two distinct primes and a repeated prime
standardFactor a number with three distinct primes or a higher multiplicity
challengeFactor a larger number with several distinct prime factors
examChoose a prime-factor form from options containing composite-factor and index-count traps

Questions about this topic

What is a prime factorisation?+

It is a product of prime numbers equal to the original whole number. Each prime is used as a factor and cannot be split into smaller whole-number factors

Why are repeated primes written as powers?+

An index records how many times the same prime is repeated, so multiplying three factors of 2 is written as two to the power of three

How can a prime-factor form be checked?+

Multiply the prime powers. The result must equal the original whole number, and every base must be prime and repeated primes must be grouped

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)