Indices

<p>The following laws hold for&nbsp;\(a,b&gt;0\) and real indices. In root notation, \(n\) is a positive integer.</p>
<p class="paragraph-header">Same base</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">\(a^m a^n=a^{m+n}\qquad\dfrac{a^m}{a^n}=a^{m-n}\qquad(a^m)^n=a^{mn}\)</p></div></div>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 540px;"><colgroup><col style="width: 100.0000%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Examples</strong></th></tr></thead><tbody><tr><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr></tbody></table></div></div>
<p class="paragraph-header">Same index</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">\(a^n b^n=(ab)^n\qquad\dfrac{a^n}{b^n}=\left(\dfrac ab\right)^n\)</p></div></div>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 540px;"><colgroup><col style="width: 100.0000%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Examples</strong></th></tr></thead><tbody><tr><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr></tbody></table></div></div>
<p class="paragraph-header">Zero, negative and fractional indices</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">\(a^0=1\qquad a^{-n}=\dfrac1{a^n}\qquad a^{\frac{m}{n}}=\sqrt[n]{a^m}\)</p></div></div>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 540px;"><colgroup><col style="width: 100.0000%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Examples</strong></th></tr></thead><tbody><tr><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr></tbody></table></div></div>
<p class="paragraph-header">Addition and subtraction</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 540px;"><colgroup><col style="width: 38.8889%;"><col style="width: 61.1111%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Skill</strong></th><th style="text-align: center; --cell-justify: center"><strong>Working</strong></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">Split shifted powers, then add like terms</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">Split, then multiply powers with the same index</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr></tbody></table></div></div>
<p>Combine like terms; index laws do not combine a sum or difference into a single power.</p>

The following laws hold for \(a,b>0\) and real indices. In root notation, \(n\) is a positive integer.

\(a^m a^n=a^{m+n}\qquad\dfrac{a^m}{a^n}=a^{m-n}\qquad(a^m)^n=a^{mn}\)

\(\dfrac{7^{3y+2}}{7^{y-1}}=7^{(3y+2)-(y-1)}=7^{2y+3}\)

\(a^n b^n=(ab)^n\qquad\dfrac{a^n}{b^n}=\left(\dfrac ab\right)^n\)

\(a^0=1\qquad a^{-n}=\dfrac1{a^n}\qquad a^{\frac{m}{n}}=\sqrt[n]{a^m}\)

\(27^{\frac{2}{3}}=\left(\sqrt[3]{27}\right)^{2}=9\)

\(q^{-3}=\dfrac{1}{q^3},\quad \dfrac{1}{x^{-6}}=x^6\)

\(\left(\dfrac{3}{5}\right)^{-x}=\left(\dfrac{5}{3}\right)^{x}\)

Split shifted powers, then add like terms

\(\begin{aligned} &3^{y+2}+3^{y+1} \\ &=(3^y)(3^2)+(3^y)(3^1) \\ &=9(3^y)+3(3^y) \\ &=12(3^y) \end{aligned}\)

Split, then multiply powers with the same index

\(\begin{aligned} &5^{y+1}\times 2^{y+2} \\ &=(5^y)(5^1)\times(2^y)(2^2) \\ &=20(5^y)(2^y) \\ &=20(10^y) \end{aligned}\)

Combine like terms; index laws do not combine a sum or difference into a single power.

Laws of Indices

Laws of Indices

Examples

Examples

Examples