Indices

<p>We just learnt to extend all the 5 laws of indices to include <strong>all integer indices. </strong></p>
<p>How about indices which are non-integer rational numbers? </p>
<p>For example, what is \({{3}^{\frac{1}{2}}}\) and \({{3}^{\frac{1}{3}}}\) equal to?</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"><p>Let \(p={{3}^{\frac{1}{2}}}\).</p></th><th style="text-align: center; --cell-justify: center"><p>Let \(p={{3}^{\frac{1}{3}}}\).</p></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center"><p>Then square both sides.</p></td><td style="text-align: center; --cell-justify: center"><p>Then cube both sides</p></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>Applying Law 3 of Indices,</p></td><td style="text-align: center; --cell-justify: center"><p>Applying Law 3 of Indices,</p></td></tr><tr><td style="text-align: center; --cell-justify: center"><p><img class="rich-text-image" src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1765782189879-tefxxyrjxy.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjU3ODIxODk4NzktdGVmeHh5cmp4eS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzMDEyMDc1LCJleHAiOjE3OTMwMTIzNzV9.HpE2oh6AXh9qttnLM6hB-N1V6-KNDA4nYr5WniU1aTc">When \(a={{b}^{n}}\), then \(b=\sqrt[n]{a}\)</p></td><td style="text-align: center; --cell-justify: center"><p><img class="rich-text-image" src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1765782189879-tefxxyrjxy.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjU3ODIxODk4NzktdGVmeHh5cmp4eS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzMDEyMDc1LCJleHAiOjE3OTMwMTIzNzV9.HpE2oh6AXh9qttnLM6hB-N1V6-KNDA4nYr5WniU1aTc">When \(a={{b}^{n}}\), then \(b=\sqrt[n]{a}\)</p></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>Hence \(p=\)</p></td><td style="text-align: center; --cell-justify: center"><p>Hence \(p=\)</p></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>This means that</p><p>\({{3}^{\frac{1}{2}}}=\)</p></td><td style="text-align: center; --cell-justify: center"><p>This means that</p><p>\({{3}^{\frac{1}{3}}}=\)</p></td></tr></tbody></table></div></div>
<p>In general, if&nbsp;\(a\) is a positive real number and \(n\) is a positive integer, we define</p>
<p style="text-align: center">\({{a}^{\frac{1}{n}}}=\sqrt[n]{a}\)</p>
<p>if&nbsp;\(a&gt;0\).</p>

We just learnt to extend all the 5 laws of indices to include **all integer indices. **

How about indices which are non-integer rational numbers? 

For example, what is \({{3}^{\frac{1}{2}}}\) and \({{3}^{\frac{1}{3}}}\) equal to?

In general, if \(a\) is a positive real number and \(n\) is a positive integer, we define

Rational Indices

Rational Indices

Video Solution:

Solution:

Solution locked

Let \(p={{3}^{\frac{1}{2}}}\).	Let \(p={{3}^{\frac{1}{3}}}\).
Then square both sides.	Then cube both sides
Applying Law 3 of Indices,	Applying Law 3 of Indices,
When \(a={{b}^{n}}\), then \(b=\sqrt[n]{a}\)	When \(a={{b}^{n}}\), then \(b=\sqrt[n]{a}\)
Hence \(p=\)	Hence \(p=\)
This means that \({{3}^{\frac{1}{2}}}=\)	This means that \({{3}^{\frac{1}{3}}}=\)