<p>Rationalisation removes surds from a denominator. The original denominator must be nonzero.</p>
<div class="content-box-placement content-box-placement--full content-box-placement--align-center" style="display:flex;justify-content:stretch;max-width:100%"><div class="content-box content-box--emphasis" data-content-box-preset="emphasis" style="box-sizing:border-box;border-color:var(--content-box-border, #1E6FBA);border-style:solid;border-width:0px;border-radius:0px;background-color:var(--content-box-background, #F5F8FC);color:var(--content-box-foreground, #111827);padding:12px 16px;font-family:'EB Garamond', serif;font-size:16px;line-height:1.35;width:100%;max-width:100%;border-left-width:3px"><p style="text-align: center">\((A+B)(A-B)=A^2-B^2\)</p></div></div>
<p class="paragraph-header">Choose a multiplier</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 570px;"><colgroup><col style="width: 26.3158%;"><col style="width: 28.0702%;"><col style="width: 45.6140%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Denominator</strong></th><th style="text-align: center; --cell-justify: center"><strong>Multiply top and bottom by</strong></th><th style="text-align: center; --cell-justify: center"><strong>Example</strong></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">Single surd \(a\sqrt b\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">Binomial \(a+h\sqrt b\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr></tbody></table></div></div>
<p class="paragraph-header">Conjugate pairs</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 480px;"><colgroup><col style="width: 50.0000%;"><col style="width: 50.0000%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Denominator</strong></th><th style="text-align: center; --cell-justify: center"><strong>Conjugate</strong></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">\(a+h\sqrt b\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">\(a\sqrt b+k\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">\(a\sqrt b+h\sqrt k\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr></tbody></table></div></div>
<p>Multiply both numerator and denominator by the conjugate. Radicands must be non-negative.</p>

Rationalisation removes surds from a denominator. The original denominator must be nonzero.

\(\begin{aligned} \dfrac{5}{2\sqrt3}&=\dfrac{5}{2\sqrt3}\times\dfrac{\sqrt3}{\sqrt3} \\ &=\dfrac{5\sqrt3}{6} \end{aligned}\)

\(\begin{aligned} \dfrac{4+\sqrt2}{3-\sqrt2}&=\dfrac{4+\sqrt2}{3-\sqrt2}\times\dfrac{3+\sqrt2}{3+\sqrt2} \\ &=\dfrac{12+4\sqrt2+3\sqrt2+2}{3^2-(\sqrt2)^2} \\ &=\dfrac{14+7\sqrt2}{7} \\ &=2+\sqrt2 \end{aligned}\)

Multiply both numerator and denominator by the conjugate. Radicands must be non-negative.

Denominator	Multiply top and bottom by	Example
Single surd \(a\sqrt b\)
Binomial \(a+h\sqrt b\)

Rationalising the Denominator

Rationalising the Denominator