Exact arithmetic with surds
A surd should be written with a rational coefficient and a square-free radical basis. Simplify the bases before adding, subtracting, multiplying or comparing exact surds
\[\sqrt{ab}=\sqrt{a}\sqrt{b}\quad(a,b\ge0)\]
- Extract square factors Write each term as a rational coefficient times a square-free radical basis \(\displaystyle \sqrt{48}=\sqrt{16\times3}=4\sqrt3\)
- Combine like radical bases Add or subtract rational coefficients only when the square-free radicands match \(\displaystyle 5\sqrt7-2\sqrt7=3\sqrt7\)
- Resimplify products Multiply coefficients and radicands, then extract square factors from the product \(\displaystyle \sqrt6\times\sqrt{15}=\sqrt{90}=3\sqrt{10}\)
- Compare positive surds exactly For positive surds, compare exact squares or certified values instead of rounded decimals \(\displaystyle 2\sqrt5<\sqrt{23}\Longleftrightarrow20<23\)
Worked example
Simplify and write the surd expression in canonical exact form
\[2\sqrt{50}-\sqrt8\]
- \(\displaystyle 2\sqrt{50}=2\sqrt{25\times2}=10\sqrt2\)Extract 25 from 50 and multiply the extracted factor by 2
- \(\displaystyle \sqrt8=\sqrt{4\times2}=2\sqrt2\)Extract 4 from 8, leaving the square-free radicand 2
- \(\displaystyle 10\sqrt2-2\sqrt2=8\sqrt2\)The radical bases match, so subtract the rational coefficients
Common mistakes
Adding radicals under one root
✕\(\displaystyle \sqrt5+\sqrt{20}=\sqrt{25}=5\)✓\(\displaystyle \sqrt5+\sqrt{20}=\sqrt5+2\sqrt5=3\sqrt5\)
Separate radicals are simplified and combined by like bases; addition is not performed inside one square rootLeaving a square factor in the radicand
✕\(\displaystyle \sqrt{72}=2\sqrt{18}\)✓\(\displaystyle \sqrt{72}=6\sqrt2\)
The radicand 18 still contains the square factor 9, so the simplification is not completeProduct not resimplified
✕\(\displaystyle \sqrt6\times\sqrt{15}=\sqrt{90}\)✓\(\displaystyle \sqrt6\times\sqrt{15}=3\sqrt{10}\)
After multiplying the radicands, extract the square factor 9 from 90Difficulty levels
introSimplify one surd or compare two positive single-term surds
standardAdd or subtract like surds, or multiply two surd terms
challengeExpand surd binomials, use conjugate products or substitute an exact surd into a quadratic
examChoose the canonical exact form and verify selected radical coefficients
Questions about this topic
What is a surd in exact form?+
A surd is an irrational radical written without a decimal approximation, usually with a rational coefficient and a square-free radicand
When can surd terms be combined?+
First simplify each term. Add or subtract coefficients only when the square-free radical bases are the same
Why must a product of radicals be simplified again?+
Multiplying radicands can create a perfect-square factor. Extract that factor so the final radical basis is square-free
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 2: Functions and graphs
In this chapter
Read restricted graphs and horizontal level setsFind the gradient of a line through two pointsFind the equation of a line through two pointsSolve a pair of simultaneous linear equationsRead the vertical-axis intercept and gradient from an equationRead the point of intersection of two linesConstruct a linear model from a descriptionInterpret the graph of a linear modelTransform a quadratic and find its key featuresDetermine a quadratic rule from exact informationAnalyse signs and solve a quadratic inequalityAnalyse a cubic graph or determine its ruleInterpret and determine a transformed power functionConnect a factorised higher-degree rule and its graphSolve a higher-degree polynomial equation or inequalityApproximate solutions from intersections of two graphsClassify a relation and state its domain and rangeEvaluate and determine function rulesApply and describe transformations of a functionAnalyse the general rational-power familyFind the maximal domain of an algebraic functionAnalyse reciprocal and square-reciprocal functionsAnalyse sideways-parabola and root-function familiesDetermine a power-family rule from features and pointsWork with sets and interval notationSimplify, compare and operate with surdsCombine function rules and ordinatesRecognise equivalent polynomial formsUse a piecewise linear modelEvaluate exact circular-function values by symmetry and periodicityKey points and features of the basic sine and cosine graphsTransform and reconstruct sine and cosine functionsFeatures and transformations of tangent functionsForm and use periodic sine and cosine modelsAnalyse exponential and logarithmic graphsInterpret and use an exponential modelSelect and interpret a function modelEvaluate and interpret a restricted rational model