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 \(\displaystyle \begin{aligned}7-12+8&=-5+8\\&=3\end{aligned}\)
Worked example
Evaluate the expression
\[(7-(2-6))\times(-2)^2\]
- \(\displaystyle \begin{aligned}&(7-(2-6))\times(-2)^2\\&=(7-(-4))\times(-2)^2\end{aligned}\)Evaluate the inner bracket
- \(\displaystyle \begin{aligned}&(7-(-4))\times(-2)^2\\&=(7+4)\times(-2)^2\end{aligned}\)Subtracting a negative adds its opposite
- \(\displaystyle \begin{aligned}&(7+4)\times(-2)^2\\&=11\times(-2)^2\end{aligned}\)Finish the outer bracket
- \(\displaystyle 11\times(-2)^2=11\times4\)Square the whole negative base
- \(\displaystyle 11\times4=44\)Multiply the two group values
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 10 Mathematics, chapter 3: Algebra and indices
In this chapter
Identify the parts of an algebraic expressionForm and evaluate algebraic expressionsSubstitute values into a formulaEvaluate an integer expressionIdentify and collect like termsMultiply and divide algebraic termsExpand brackets and simplifyFind the HCF and factorise algebraic expressionsSimplify, multiply and divide algebraic fractionsAdd and subtract algebraic fractionsIndex notation and index laws 1 and 2Index laws 3 to 5 and the zero powerNegative indices and combined index lawsScientific notationRound to decimal places or significant figuresScientific notation with significant figuresExponential growth, decay and compound interestModel a compound-interest investment or loan with a recurrence relationFind term n of a geometric sequence using the ruleInterpret and use logarithmic scalesImplement algorithms with lists