Signs, groups and order
Work within each bracket and root group. Keep the rest of the expression visible as you simplify one part
- Subtract the whole group Find the group's value before subtracting it \(\displaystyle \begin{aligned}11-(4-10)&=11-(-6)\\&=11+6\\&=17\end{aligned}\)
- Powers and roots A square uses two equal factors and a cube uses three. Square roots of perfect squares are non-negative; cube roots of negative cubes are negative \(\displaystyle \begin{aligned}(-6)^2&=36\\(-3)^3&=-27\\\sqrt{196}&=14\\\sqrt[3]{-216}&=-6\end{aligned}\)
- Multiply and divide from the left These operations share priority. Matching signs give a positive result; different signs give a negative result \(\displaystyle \begin{aligned}-54\div9\times(-4)&=-6\times(-4)\\&=24\end{aligned}\)
- Add and subtract from the left Do these after powers, roots, multiplication and division within the same group \(\displaystyle \begin{aligned}7-12+8&=-5+8\\&=3\end{aligned}\)
Common mistakes
Losing the bracket
✕\(\displaystyle 12-(5-9)=12-5-9\)✓\(\displaystyle \begin{aligned}12-(5-9)&=12-(-4)\\&=16\end{aligned}\)
The outer subtraction applies to the entire differenceMoving the minus into the base
✕\(\displaystyle -8^2=64\)✓\(\displaystyle -8^2=-64\)
The power applies to the positive base unless brackets include the minus signTaking roots before adding
✕\(\displaystyle \sqrt{56+25}=\sqrt{56}+\sqrt{25}\)✓\(\displaystyle \begin{aligned}\sqrt{56+25}&=\sqrt{81}\\&=9\end{aligned}\)
The root bar groups the whole sum. A square root does not distribute over additionDifficulty levels
introEvaluate a square, cube or exact root, then combine it with signed numbers
standardEvaluate bracketed differences with integers, powers or roots
challengeCombine groups and products, sometimes with both a power and a root
examChoose an integer result from options built from order and sign errors
Questions about this topic
Must every bracket be done before any power?+
Finish a group before using it. Separate groups can be simplified in either order
Why can a negative base have a positive square?+
Its two negative factors multiply to a positive result. A leading minus outside the square is a different expression
What does evaluate mean?+
Find the number equal to the expression
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 9 Mathematics, chapter 1: Number and financial mathematics
In this chapter
Evaluate an integer expressionFactors, multiples, HCF, LCM and primesRound to decimal places or significant figuresEstimate an expression using rounded inputsConvert between fractions, mixed numbers and decimalsClassify real numbers and compare exact valuesAdd, subtract, multiply and divide fractions and mixed numbersWrite a ratio from countsWrite ratios in simplest formComplete equivalent ratiosUse a constant rateDivide a quantity in a given ratioDecide which option is the better valueConvert between fractions, decimals and percentagesFind a percentage of a quantityExpress one quantity as a percentage of another1F Percentages and money: Find the original amount from a percentageIncrease or decrease a quantity by a percentageFind percentage changePercentage error against a reference value1G Percentage change and percentage error: Find the original amount from a percentageProfit, mark-up and discountIncome and taxSimple interestCompound interest (optional extension)