e39465ad2f
new file: Files/flashplayer_32_sa.exe
new file: favicon.ico
new file: globe.gif
new file: imgs/download.png
new file: imgs/zuck.jpg
new file: index.html
new file: other.ico
new file: script.js
new file: site.webmanifest
new file: sitemap.html
new file: styles/backround.css
new file: styles/border.css
new file: styles/fonts/Titillium_Web/OFL.txt
new file: styles/fonts/Titillium_Web/TitilliumWeb-Black.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Bold.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-BoldItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-ExtraLight.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-ExtraLightItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Italic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Light.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-LightItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Regular.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-SemiBold.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-SemiBoldItalic.ttf
new file: styles/fonts/webfontkit-20221027-163353/generator_config.txt
new file: styles/fonts/webfontkit-20221027-163353/specimen_files/grid_12-825-55-15.css
new file: styles/fonts/webfontkit-20221027-163353/specimen_files/specimen_stylesheet.css
new file: styles/fonts/webfontkit-20221027-163353/stylesheet.css
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-demo.html
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-webfont.woff
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-webfont.woff2
new file: styles/fonts/webfontkit-20221027-165950/generator_config.txt
new file: styles/fonts/webfontkit-20221027-165950/specimen_files/grid_12-825-55-15.css
new file: styles/fonts/webfontkit-20221027-165950/specimen_files/specimen_stylesheet.css
new file: styles/fonts/webfontkit-20221027-165950/stylesheet.css
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-demo.html
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-webfont.woff
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-webfont.woff2
new file: styles/style.css
new file: tools/2048/.gitignore
new file: tools/2048/.jshintrc
new file: tools/2048/CONTRIBUTING.md
new file: tools/2048/LICENSE.txt
new file: tools/2048/README.md
new file: tools/2048/Rakefile
new file: tools/2048/favicon.ico
new file: tools/2048/index.html
new file: tools/2048/js/animframe_polyfill.js
new file: tools/2048/js/application.js
new file: tools/2048/js/bind_polyfill.js
new file: tools/2048/js/classlist_polyfill.js
new file: tools/2048/js/game_manager.js
new file: tools/2048/js/grid.js
new file: tools/2048/js/html_actuator.js
new file: tools/2048/js/keyboard_input_manager.js
new file: tools/2048/js/local_storage_manager.js
new file: tools/2048/js/tile.js
new file: tools/2048/meta/apple-touch-icon.png
new file: tools/webretro/cores/neocd_libretro.js
new file: tools/webretro/cores/neocd_libretro.wasm
new file: tools/webretro/cores/nestopia_libretro.js
new file: tools/webretro/cores/nestopia_libretro.wasm
new file: tools/webretro/cores/o2em_libretro.js
new file: tools/webretro/cores/o2em_libretro.wasm
new file: tools/webretro/cores/opera_libretro.js
new file: tools/webretro/cores/opera_libretro.wasm
232 lines
9.9 KiB
HTML
232 lines
9.9 KiB
HTML
<!doctype html>
|
|
<html lang="en"><!-- #BeginTemplate "/Templates/Main.dwt" --><!-- DW6 -->
|
|
|
|
<!-- Mirrored from www.mathsisfun.com/algebra/negative-exponents.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:48:14 GMT -->
|
|
<head>
|
|
<!-- #BeginEditable "doctitle" -->
|
|
<title>Negative Exponents</title>
|
|
<script language="JavaScript" type="text/javascript">reSpell=[["favorite","favourite"]];</script>
|
|
<!-- #EndEditable -->
|
|
<meta name="keywords" content="math, maths, mathematics, school, homework, education">
|
|
<meta http-equiv="content-type" content="text/html; charset=utf-8">
|
|
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
|
|
<meta name="HandheldFriendly" content="true">
|
|
<meta http-equiv="pics-label" content='(PICS-1.1 "http://www.classify.org/safesurf/" L gen true for "http://www.mathsisfun.com" r (SS~~000 1))'>
|
|
<link rel="stylesheet" type="text/css" href="../style3.css" />
|
|
<script src="../main3.js" type="text/javascript"></script>
|
|
</head>
|
|
|
|
<body id="bodybg">
|
|
<div class="bg">
|
|
<div id="stt"></div>
|
|
<div id="hdr"></div>
|
|
<div id="logo"><a href="../index.html"><img src="../images/style/logo.svg" alt="Math is Fun" /></a></div>
|
|
<div id="gtran"><script type="text/javascript">document.write(getTrans());</script></div>
|
|
<div id="gplus"><script type="text/javascript">document.write(getGPlus());</script></div>
|
|
<div id="adTopOuter" class="centerfull noprint">
|
|
<div id="adTop">
|
|
<script type="text/javascript">document.write(getAdTop());</script>
|
|
</div>
|
|
</div>
|
|
<div id="adHide">
|
|
<div id="showAds1"><a href="javascript:showAds()">Show Ads</a></div>
|
|
<div id="hideAds1"><a href="javascript:hideAds()">Hide Ads</a><br>
|
|
<a href="../about-ads.html">About Ads</a></div>
|
|
</div>
|
|
<div id="menuWide" class="menu">
|
|
<script type="text/javascript">document.write(getMenu(0));</script>
|
|
</div>
|
|
<div id="linkto">
|
|
<div id="linktort"><script type="text/javascript">document.write(getLinks());</script></div>
|
|
</div>
|
|
<div id="search" role="search"><script type="text/javascript">document.write(getSearch());</script></div>
|
|
<div id="menuSlim" class="menu">
|
|
<script type="text/javascript">document.write(getMenu(1));</script>
|
|
</div>
|
|
<div id="menuTiny" class="menu">
|
|
<script type="text/javascript">document.write(getMenu(2));</script>
|
|
</div>
|
|
<div id="extra"></div>
|
|
</div>
|
|
<div id="content" role="main"><!-- #BeginEditable "Body" -->
|
|
<h1 align="center">Negative Exponents</h1>
|
|
<p align="center"><a href="../exponent.html">Exponents</a> are also called <b>Powers</b> or <b>Indices</b></p>
|
|
<p>Let us first look at what an "exponent" is:</p>
|
|
<div class="simple">
|
|
<table border="0" cellpadding="5" align="center">
|
|
<tr>
|
|
<td><img src="images/exponent-8-2.svg" alt="8 to the Power 2" /></td>
|
|
<td>
|
|
<p>The <b>exponent</b> of a number says <b>how many times </b>to use<br>
|
|
the number in a multiplication.</p>
|
|
<p class="Larger">In this example: <b>8<sup>2</sup> = 8 × 8 = 64</b><br />
|
|
</p></td>
|
|
</tr>
|
|
<tr>
|
|
<td colspan="2">In words: 8<sup>2</sup> can be called "8 to the second power", "8 to the power 2" <br>
|
|
or
|
|
simply "8 squared"</td>
|
|
</tr>
|
|
</table>
|
|
</div>
|
|
|
|
<p> </p>
|
|
<div class="example">
|
|
<h3 class="Larger">Example: <b>5<sup>3</sup> = 5 × 5 × 5 = 125</b></h3>
|
|
<p class="Larger">In words: 5<sup>3</sup> can be called "5 to the third power", "5 to the power 3" or simply
|
|
"5 cubed"</p>
|
|
</div>
|
|
<p class="Larger"><b>In general</b>:</p>
|
|
<table border="0" align="center">
|
|
<tr>
|
|
<td align="right"><span class="large">a<sup>n</sup></span> tells you to use <b>a</b> in a multiplication
|
|
<b>n</b> times:</td>
|
|
<td align="right"> </td>
|
|
<td><img src="images/exponent-definition.gif" alt="exponent definition" width="197" height="68" /></td>
|
|
</tr>
|
|
</table>
|
|
<p>But those are <b>positive exponents</b>, what about something like:</p>
|
|
<p align="center"><span class="huge">8<sup>-2</sup></span></p>
|
|
<p align="center" class="larger">That exponent is <b>negative</b> ... what does it mean?</p>
|
|
<h2>Negative Exponents</h2>
|
|
<p>Negative? What could be the opposite of multiplying? <b>Dividing! </b></p>
|
|
<div class="def">
|
|
<p align="center" class="larger"><b>Dividing</b> is the <a href="../numbers/inverse.html">inverse</a> (opposite) of <b>Multiplying</b>.</p>
|
|
</div>
|
|
<p>A negative exponent means how many times<b> to
|
|
divide</b> by the number. </p>
|
|
<div class="example">
|
|
<p class="larger">Example: <b>8<sup>-1</sup> = 1 ÷ 8 = 1/8 = 0.125</b></p>
|
|
</div>
|
|
<p>Or many divides:</p>
|
|
<div class="example">
|
|
<p class="larger">Example: <b>5<sup>-3</sup> = 1 ÷ 5 ÷ 5 ÷ 5 = 0.008</b></p>
|
|
</div>
|
|
<p>But that can be done an easier way:</p>
|
|
|
|
|
|
|
|
<div class="example">
|
|
<p class="larger"><b>5<sup>-3</sup></b> could also be calculated like:</p>
|
|
<p align="center"> <span class="larger"><b>1 ÷ (5 × 5 × 5) = 1/5<sup>3</sup> = 1/125 = 0.008</b> </span></p>
|
|
</div>
|
|
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/negative-exponent.gif" alt="negative-exponent" width="99" height="57" /></p>
|
|
<p>That last example showed an easier way to handle negative exponents:</p>
|
|
<ul>
|
|
<li>Calculate the positive exponent (<span class="large">a<sup>n</sup></span>)</li>
|
|
<li>Then take the <a href="../reciprocal.html">Reciprocal</a> (i.e. <span class="large">1/a<sup>n</sup></span>) </li>
|
|
</ul>
|
|
<p> </p>
|
|
<div class="def">
|
|
<p align="center" class="larger">To change the <b>sign</b> (plus to minus, or minus to plus) of the <b>exponent</b>,<br />
|
|
use the <a href="../reciprocal.html">Reciprocal</a> (i.e. <span class="large">1/a<sup>n</sup></span>)</p>
|
|
</div>
|
|
<p align="center" class="larger"> </p>
|
|
<p class="larger">So, what about <b>8<sup>-2</sup></b> ?</p>
|
|
<div class="example">
|
|
<p class="larger">Example: <b>8<sup>-2</sup> = 1 ÷ 8 ÷ 8 = 1/8<sup>2</sup> = 1/64 = 0.015625</b></p>
|
|
</div>
|
|
<p> </p>
|
|
<p>More Examples:</p>
|
|
<div class="beach">
|
|
<table width="100%" border="0" align="center">
|
|
<tr>
|
|
<th>Negative Exponent</th>
|
|
<th> </th>
|
|
<th>Reciprocal of <br>
|
|
Positive Exponent</th>
|
|
<th> </th>
|
|
<th>Answer</th>
|
|
</tr>
|
|
<tr align="center">
|
|
<td><b>4<sup>-2</sup></b></td>
|
|
<td> = </td>
|
|
<td><b>1 / 4<sup>2</sup> </b></td>
|
|
<td> = </td>
|
|
<td><b>1/16 = 0.0625</b></td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td><b>10<sup>-3</sup></b></td>
|
|
<td> = </td>
|
|
<td><b>1 / 10<sup>3</sup> </b></td>
|
|
<td> = </td>
|
|
<td><b>1/1,000 = 0.001</b></td>
|
|
</tr>
|
|
</table>
|
|
</div>
|
|
<h2>It All Makes Sense</h2>
|
|
<p>My favorite method is to start with "1" and then multiply or divide as many times as the exponent says, then you will get the right answer, for example:</p>
|
|
<div class="simple">
|
|
<table border="0" align="center">
|
|
<tr align="center">
|
|
<th colspan="4">Example: Powers of 5</th>
|
|
</tr>
|
|
<tr>
|
|
<td width="60" align="center"> </td>
|
|
<td width="90" align="center">.. etc..</td>
|
|
<td width="60" align="center"> </td>
|
|
<td rowspan="7"><img src="images/larger-smaller-5.svg" alt="exponent larger smaller" /></td>
|
|
</tr>
|
|
<tr>
|
|
<td><b>5<sup>2</sup></b></td>
|
|
<td><b>1 × 5 × 5</b></td>
|
|
<td>25</td>
|
|
</tr>
|
|
<tr>
|
|
<td><b>5<sup>1</sup></b></td>
|
|
<td><b>1 × 5</b></td>
|
|
<td>5</td>
|
|
</tr>
|
|
<tr>
|
|
<td><b>5<sup>0</sup></b></td>
|
|
<td><b>1</b></td>
|
|
<td>1</td>
|
|
</tr>
|
|
<tr>
|
|
<td><b>5<sup>-1</sup></b></td>
|
|
<td><b>1 ÷ 5</b></td>
|
|
<td>0.2</td>
|
|
</tr>
|
|
<tr>
|
|
<td><b>5<sup>-2</sup></b></td>
|
|
<td><b>1 ÷ 5 ÷ 5</b></td>
|
|
<td>0.04</td>
|
|
</tr>
|
|
<tr>
|
|
<td align="center"> </td>
|
|
<td align="center">.. etc..</td>
|
|
<td align="center"> </td>
|
|
</tr>
|
|
</table>
|
|
</div>
|
|
<p>If you look at that table, you will see that positive, zero or negative exponents are really part of the same (fairly simple) pattern.</p>
|
|
<p> </p>
|
|
<div class="questions">
|
|
|
|
<script type="text/javascript">getQ(751, 2024, 2025, 2026, 752, 2027, 3147, 3148, 3149, 3150);</script>
|
|
|
|
</div>
|
|
|
|
<div class="related">
|
|
<a href="exponent-laws.html">Laws of Exponents</a>
|
|
<a href="exponent-fractional.html">Fractional Exponents</a>
|
|
<a href="../index-notation-powers.html">Powers of 10</a>
|
|
<a href="../decimals.html">Decimals</a>
|
|
<a href="../metric-numbers.html">Metric Numbers</a></div>
|
|
<!-- #EndEditable --></div>
|
|
<div id="adend" class="centerfull noprint">
|
|
<script type="text/javascript">document.write(getAdEnd());</script>
|
|
</div>
|
|
<div id="footer" class="centerfull noprint">
|
|
<script type="text/javascript">document.write(getFooter());</script>
|
|
</div>
|
|
<div id="copyrt">
|
|
Copyright © 2017 MathsIsFun.com
|
|
</div>
|
|
|
|
<script type="text/javascript">document.write(getBodyEnd());</script>
|
|
</body>
|
|
<!-- #EndTemplate -->
|
|
<!-- Mirrored from www.mathsisfun.com/algebra/negative-exponents.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:48:14 GMT -->
|
|
</html>
|