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Rationalise a surd denominator

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Rationalise a surd denominator

Rationalise a denominator by multiplying the numerator and denominator by a suitable factor. Use a conjugate for a binomial and simplify the exact result afterwards

\[\frac{a}{b}=\frac{a\times k}{b\times k}\]
  1. 1Multiply top and bottom by the same factor Multiplying the numerator and denominator by the same non-zero factor preserves the value \(\displaystyle \frac{a}{b}=\frac{a\times k}{b\times k}\)
  2. 2Use the radical for a monomial For a denominator with one square root, multiply by that same square root \(\displaystyle \frac{1}{\sqrt{3}}\times\frac{\sqrt{3}}{\sqrt{3}}=\frac{\sqrt{3}}{3}\)
  3. 3Use the sign-switched conjugate For a binomial denominator, switch the sign between its rational and surd terms \(\displaystyle \left(5+2\sqrt{3}\right)\left(5-2\sqrt{3}\right)=25-12=13\)
  4. 4Reduce the final exact form Simplify numerical factors and remove any square factors from a surd radicand \(\displaystyle \frac{3\sqrt{2}}{6}=\frac{\sqrt{2}}{2}\)
Year 11, VCE Units 1 & 2, Chapter 2, 4 difficulty levels, No calculatorBuild a worksheet

Rationalising surd denominators

Rationalise a denominator by multiplying the numerator and denominator by a suitable factor. Use a conjugate for a binomial and simplify the exact result afterwards

\[\frac{a}{b}=\frac{a\times k}{b\times k}\]
  1. Multiply top and bottom by the same factor Multiplying the numerator and denominator by the same non-zero factor preserves the value \(\displaystyle \frac{a}{b}=\frac{a\times k}{b\times k}\)
  2. Use the radical for a monomial For a denominator with one square root, multiply by that same square root \(\displaystyle \frac{1}{\sqrt{3}}\times\frac{\sqrt{3}}{\sqrt{3}}=\frac{\sqrt{3}}{3}\)
  3. Use the sign-switched conjugate For a binomial denominator, switch the sign between its rational and surd terms \(\displaystyle \left(5+2\sqrt{3}\right)\left(5-2\sqrt{3}\right)=25-12=13\)
  4. Reduce the final exact form Simplify numerical factors and remove any square factors from a surd radicand \(\displaystyle \frac{3\sqrt{2}}{6}=\frac{\sqrt{2}}{2}\)

Worked example

Rationalise and simplify the denominator in the fraction below

\[\frac{3}{4+\sqrt{7}}\]
  1. \(\displaystyle \frac{3}{4+\sqrt{7}}\times\frac{4-\sqrt{7}}{4-\sqrt{7}}\)Multiply top and bottom by the conjugate of the denominator
  2. \(\displaystyle \left(4+\sqrt{7}\right)\left(4-\sqrt{7}\right)=4^{2}-\left(\sqrt{7}\right)^{2}=16-7=9\)The conjugate product removes the surd from the denominator
  3. \(\displaystyle 3\left(4-\sqrt{7}\right)=12-3\sqrt{7}\)Expand the numerator after applying the same conjugate factor
  4. \(\displaystyle \frac{12-3\sqrt{7}}{9}=\frac{4-\sqrt{7}}{3}\)Divide each numerator term by 9 and reduce the common factor 3

Common mistakes

Not switching the conjugate sign
✕\(\displaystyle \left(5+\sqrt{2}\right)\left(5+\sqrt{2}\right)=27+10\sqrt{2}\)✓\(\displaystyle \left(5+\sqrt{2}\right)\left(5-\sqrt{2}\right)=23\)
The conjugate changes the sign of the radical term, so the cross terms cancel
Applying the factor to the denominator only
✕\(\displaystyle \frac{3}{\sqrt{5}}=\frac{3}{5}\)✓\(\displaystyle \frac{3}{\sqrt{5}}=\frac{3\sqrt{5}}{5}\)
The same factor must be applied to the numerator and denominator to preserve the fraction's value
Leaving a square factor in a radicand
✕\(\displaystyle \sqrt{12}\)✓\(\displaystyle \sqrt{12}=2\sqrt{3}\)
Extract the greatest square factor from the radicand before finalising the exact form

Difficulty levels

introThis tier rationalises denominators containing one square root
standardThis tier uses conjugates for binomial surd denominators
challengeThis tier simplifies surd numerators, outside factors and collected exact forms
examThis tier identifies the rationalising multiplier and gives the final simplified form

Questions about this topic

What is a conjugate in surd rationalisation?+

The conjugate keeps the rational term and switches the sign of the surd term in a binomial

Why do the cross terms disappear when conjugates are multiplied?+

The two cross terms have equal magnitude and opposite signs, leaving a difference of squares

What is a rational denominator?+

It is a denominator that contains no surd or other irrational term

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, Mathematical Methods, chapter 2: Algebraic foundations

In this chapter

Expand binomial and special productsFactorise by common factors and difference of perfect squaresFactorise monic and non-monic quadratic trinomialsFactorise four-term expressions by groupingRecognise equivalent polynomial formsSimplify and operate on algebraic fractionsSimplify and combine algebraic fractionsEvaluate factorial and combination expressionsPascal rows and binomial coefficient structureLocate and evaluate a term in a binomial expansionClassify real numbers and recognise undefined formsWork with sets and interval notationSimplify, compare and operate with surdsRationalise a surd denominator