Simplifying terms and sums
Choose the rule from the operation. Keep signs attached to terms and keep all denominator restrictions after cancelling
- Products Multiply coefficients and add exponents for matching bases. The displayed example uses positive integer exponents \(\displaystyle 3z^{2}\times(-5z)=-15z^{3}\)
- Quotients Divide coefficients and cancel matching factors. Cancelling a nonzero factor completely leaves exponent zero, so its value is one. Omit that factor, but retain its nonzero restriction \(\displaystyle \begin{gathered}\frac{16a^{3}b}{-8ab}=-2a^{2}\\\quad(a\ne0,\ b\ne0)\end{gathered}\)
- Like terms The letters and powers must match. Their order within a product does not matter. If signed coefficients add to zero, the whole term is zero and can be omitted \(\displaystyle 7jk^{2}-4k^{2}j=3jk^{2}\)
- Unlike terms Keep different powers and constants in their own groups \(\displaystyle \begin{gathered}6v^{2}+2v-3v^{2}+9\\=3v^{2}+2v+9\end{gathered}\)
Worked example
Simplify the quotient. Both pronumerals are nonzero
\[\frac{42h^{3}k^{2}}{-6hk}\]
- \(\displaystyle =-7\frac{h^{3}k^{2}}{hk}\)Divide the signed coefficients and carry both pronumeral factors through
- \(\displaystyle =-7h^{2}\frac{k^{2}}{k}\)Cancel one matching factor of the first pronumeral
- \(\displaystyle =-7h^{2}k\)Cancel one matching factor of the second pronumeral. The original nonzero restrictions still apply
Common mistakes
Adding coefficients in a product
✕\(\displaystyle (-4d)\times(6de)=2d^{2}e\)✓\(\displaystyle (-4d)\times(6de)=-24d^{2}e\)
A product requires multiplication of the signed coefficientsAdding powers in a quotient
✕\(\displaystyle \frac{u^{6}}{u^{2}}=u^{8}\)✓\(\displaystyle \frac{u^{6}}{u^{2}}=u^{4}\quad(u\ne0)\)
For a nonzero base, cancel the matching factors by subtracting the denominator exponentLosing a sign when collecting
✕\(\displaystyle \begin{gathered}5s^{2}-8s+3s^{2}+2s\\=8s^{2}+10s\end{gathered}\)✓\(\displaystyle \begin{gathered}5s^{2}-8s+3s^{2}+2s\\=8s^{2}-6s\end{gathered}\)
Reordering terms does not change their signs. Add the linear coefficients with their signsDifficulty levels
introRecognise simple products, quotients and short collections with positive coefficients
standardUse signed coefficients, reordered factors and cancellation. Calculate the requested coefficients or powers
challengeCollect separated groups with different powers, or cancel matching factors in quotients, including higher powers and complete cancellation
examRecognise simplified results for standard and challenge structures without extra scaffolding
Questions about this topic
When are terms alike?+
Their pronumeral factors and powers match. The order of multiplied letters does not matter
Why must a denominator be nonzero?+
Division by zero is undefined. Cancelling factors does not remove the original restriction
Does a matching value prove two expressions are equivalent?+
No. Equivalent expressions agree for every allowed value. One value giving different results is enough to show that they are not equivalent
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 9 Mathematics, chapter 2: Linear equations
In this chapter
Translate between a description and an algebraic expressionEvaluate an algebraic expression by substitutionTranslate between a description and an algebraic expressionSimplify algebraic terms and collect like termsSimplify algebraic terms and collect like termsExpand brackets and collect like terms2D Two-step linear equations: Solve a linear equation2E Linear equations with brackets: Solve a linear equationSet up and solve a practical linear equationSolve and interpret a linear inequationRepresent and read inequalities on a number lineSubstitute values into a formulaSolve for an unknown or recognise a rearranged formulaSolve a pair of simultaneous linear equations (optional extension)Solve a pair of simultaneous linear equations by elimination (optional extension)Set up a pair of simultaneous equations from a description (optional extension)Solve a practical problem with simultaneous equations (optional extension)