<p>A surd is an irrational root written in exact form.</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">\(\sqrt{ab}=\sqrt a\sqrt b\quad(a,b\ge0)\)</p></div></div>
<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">\(\sqrt{\dfrac ab}=\dfrac{\sqrt a}{\sqrt b}\quad(a\ge0,\ b&gt;0)\)</p></div></div>
<p class="paragraph-header">Simplifying surds</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 540px;"><colgroup><col style="width: 50.0000%;"><col style="width: 50.0000%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Operation</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">Addition and subtraction</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">Multiplication</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">Division</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr></tbody></table></div></div>
<p class="paragraph-header">Expanding brackets</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 560px;"><colgroup><col style="width: 42.8571%;"><col style="width: 57.1429%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Identity</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">\((A+B)^2=A^2+2AB+B^2\)</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">\((A+B)(A-B)=A^2-B^2\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr></tbody></table></div></div>
<p>Conjugates have opposite signs between their terms: \(a+\sqrt b\) and \(a-\sqrt b\).</p>
<p class="paragraph-header">Simplest and entire forms</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 560px;"><colgroup><col style="width: 26.7857%;"><col style="width: 37.5000%;"><col style="width: 35.7143%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Task</strong></th><th style="text-align: center; --cell-justify: center"><strong>Method</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">Simplest form \(a\sqrt b\)</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">Entire surd \(\sqrt p\)</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>Roots do not split over sums or differences: \(\sqrt{a+b}\ne\sqrt a+\sqrt b\) in general.</p>

A surd is an irrational root written in exact form.

\(\sqrt{ab}=\sqrt a\sqrt b\quad(a,b\ge0)\)

\(\sqrt{\dfrac ab}=\dfrac{\sqrt a}{\sqrt b}\quad(a\ge0,\ b>0)\)

\(4\sqrt5+3\sqrt5=7\sqrt5\)
\(2\sqrt3+5\sqrt7\): unlike surds, cannot combine

\(\begin{aligned} 3\sqrt2\times5\sqrt2&=15(2)=30 \\ 2\sqrt3\times4\sqrt5&=8\sqrt{15} \end{aligned}\)

\(\dfrac{9\sqrt{22}}{3\sqrt{11}}=3\sqrt{2}\)

\(\begin{aligned} (3\sqrt2+1)^2&=(3\sqrt2)^2+2(3\sqrt2)+1^2 \\ &=18+6\sqrt2+1 \\ &=19+6\sqrt2 \end{aligned}\)

\(\begin{aligned} (2\sqrt5+3)(2\sqrt5-3)&=(2\sqrt5)^2-3^2 \\ &=20-9 \\ &=11 \end{aligned}\)

Conjugates have opposite signs between their terms: \(a+\sqrt b\) and \(a-\sqrt b\).

Take out the largest square factor of the radicand.

\(\begin{aligned} \sqrt{180}&=\sqrt{36\times5} \\ &=6\sqrt5 \end{aligned}\)

Bring the coefficient under the root sign.

\(\begin{aligned} 5\sqrt2&=\sqrt{25}\times\sqrt2 \\ &=\sqrt{50} \end{aligned}\)

Roots do not split over sums or differences: \(\sqrt{a+b}\ne\sqrt a+\sqrt b\) in general.

Task	Method	Example
Simplest form \(a\sqrt b\)
Entire surd \(\sqrt p\)

Law of Surds

Law of Surds

Operation	Example
Addition and subtraction
Multiplication
Division

Identity	Example
\((A+B)^2=A^2+2AB+B^2\)
\((A+B)(A-B)=A^2-B^2\)