Indices

<p class="paragraph-header">Fractional indices and roots</p>
<p>For&nbsp;\(a&gt;0\) and positive integers \(m\) and \(n\), \(a^{\frac1n}=\sqrt[n]{a}\) and \(a^{\frac mn}=(\sqrt[n]{a})^m\). Take the root first when it gives a small whole number.</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>Power</strong></th><th style="text-align: center; --cell-justify: center"><strong>Meaning</strong></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">\(16^{\frac34}\)</td><td style="text-align: center; --cell-justify: center">\((\sqrt[4]{16})^3=2^3=8\)</td></tr><tr><td style="text-align: center; --cell-justify: center">\(16^{-\frac34}\)</td><td style="text-align: center; --cell-justify: center">\(\dfrac1{16^{\frac34}}=\dfrac18\)</td></tr></tbody></table></div></div>
<p>Use the positive root for these positive bases. For example, \(16^{\frac12}=4\); it is not \(\pm4\). The equation \(x^2=16\) is a different task and has two solutions.</p>

For \(a>0\) and positive integers \(m\) and \(n\), \(a^{\frac1n}=\sqrt[n]{a}\) and \(a^{\frac mn}=(\sqrt[n]{a})^m\). Take the root first when it gives a small whole number.

Use the positive root for these positive bases. For example, \(16^{\frac12}=4\); it is not \(\pm4\). The equation \(x^2=16\) is a different task and has two solutions.

Power	Meaning
\(16^{\frac34}\)	\((\sqrt[4]{16})^3=2^3=8\)
\(16^{-\frac34}\)	\(\dfrac1{16^{\frac34}}=\dfrac18\)