Indices

<p class="paragraph-header">Index laws and reciprocals</p>
<p>A positive integer index counts equal factors: \(a^3=a\times a\times a\). In the rules below, bases in denominators are nonzero.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Rule</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">\(a^m a^n=a^{m+n}\)</td><td style="text-align: center; --cell-justify: center">\(x^3x^4=x^7\)</td></tr><tr><td style="text-align: center; --cell-justify: center">\(\dfrac{a^m}{a^n}=a^{m-n}\)</td><td style="text-align: center; --cell-justify: center">\(\dfrac{x^6}{x^2}=x^4\)</td></tr><tr><td style="text-align: center; --cell-justify: center">\((a^m)^n=a^{mn}\)</td><td style="text-align: center; --cell-justify: center">\((x^3)^4=x^{12}\)</td></tr><tr><td style="text-align: center; --cell-justify: center">\((ab)^n=a^nb^n\)</td><td style="text-align: center; --cell-justify: center">\((2x)^3=8x^3\)</td></tr><tr><td style="text-align: center; --cell-justify: center">\(a^0=1\), for&nbsp;\(a\ne0\)</td><td style="text-align: center; --cell-justify: center">\(7^0=1\)</td></tr><tr><td style="text-align: center; --cell-justify: center">\(a^{-n}=\dfrac1{a^n}\)</td><td style="text-align: center; --cell-justify: center">\(2^{-3}=\dfrac18\)</td></tr></tbody></table></div></div>
<p>A negative index means a reciprocal, not a negative value. Add indices only when multiplying powers with the same base; \(a^m+a^n\) is a sum and needs a different step.</p>

A positive integer index counts equal factors: \(a^3=a\times a\times a\). In the rules below, bases in denominators are nonzero.

A negative index means a reciprocal, not a negative value. Add indices only when multiplying powers with the same base; \(a^m+a^n\) is a sum and needs a different step.

Rule	Example
\(a^m a^n=a^{m+n}\)	\(x^3x^4=x^7\)
\(\dfrac{a^m}{a^n}=a^{m-n}\)	\(\dfrac{x^6}{x^2}=x^4\)
\((a^m)^n=a^{mn}\)	\((x^3)^4=x^{12}\)
\((ab)^n=a^nb^n\)	\((2x)^3=8x^3\)
\(a^0=1\), for \(a\ne0\)	\(7^0=1\)
\(a^{-n}=\dfrac1{a^n}\)	\(2^{-3}=\dfrac18\)