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<!DOCTYPE html>
<html data-require="math math-format">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Simplifying radicals</title>
<script data-main="../local-only/main.js" src="../local-only/require.js"></script>
</head>
<body>
<div class="exercise">
<div class="problems">
<div id="already-simplified" data-weight="2">
<div class="vars">
<var id="NUM">randFromArray([30, 42, 66, 70, 105, 110, 154, 165, 210])</var>
<var id="FACTORS">getPrimeFactorization(NUM).join("\\cdot")</var>
</div>
<p class="question">Simplify <code>\sqrt{<var>NUM</var>}</code>.</p>
<div class="solution" data-type="radical"><var>NUM</var></div>
<div class="hints">
<p><code><var>NUM</var> = <var>FACTORS</var></code></p>
<p><code><var>NUM</var></code> has no perfect-square factors, so <code>\sqrt{<var>NUM</var>}</code> is already the simplest form.</p>
<p><code>\sqrt{<var>NUM</var>} = 1\sqrt{<var>NUM</var>}</code> or just <code>\sqrt{<var>NUM</var>}</code>.</p>
</div>
</div>
<div id="perfect-squares" data-weight="2">
<div class="vars">
<var id="NUM">randFromArray([9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225])</var>
<var id="SPLIT">splitRadical(NUM)</var>
<var id="COEFFICIENT">SPLIT[0] === 1 ? "" : SPLIT[0]</var>
</div>
<p class="question">Simplify <code>\sqrt{<var>NUM</var>}</code>.</p>
<div class="solution" data-type="radical"><var>NUM</var></div>
<div class="hints">
<p><code><var>NUM</var> = <var>round(sqrt(NUM))</var>^2</code></p>
<p>So, <code>\sqrt{<var>NUM</var>} = \sqrt{<var>SPLIT[0]</var>^2} = <var>SPLIT[0]</var></code></p>
<p><code>\sqrt{<var>NUM</var>} = <var>SPLIT[0]</var>\sqrt{1}</code> or just <code><var>SPLIT[0]</var></code>.</p>
</div>
</div>
<div id="not-yet-simple" data-weight="6">
<div class="vars">
<var id="NUM">randFromArray([8, 12, 18, 20, 24, 27, 28, 32, 40, 44, 45, 48, 50, 54, 56, 60, 63, 72, 75, 80, 88, 90, 98, 99, 120, 125, 128, 140, 150, 160, 175, 180, 200, 216])</var>
<var id="SPLIT">splitRadical(NUM)</var>
<var id="COEFFICIENT">SPLIT[0] === 1 ? "" : SPLIT[0]</var>
<var id="RADICAL">SPLIT[1] === 1 ? "" : SPLIT[1]</var>
</div>
<p class="question">Simplify <code>\sqrt{<var>NUM</var>}</code>.</p>
<div class="solution" data-type="radical"><var>NUM</var></div>
<div class="hints">
<p>The largest perfect square that divides <code><var>NUM</var></code> is <code><var>COEFFICIENT * COEFFICIENT</var></code>.</p>
<p><code><var>NUM</var> = <var>COEFFICIENT * COEFFICIENT</var> \cdot <var>RADICAL</var></code></p>
<p><code>\sqrt{<var>NUM</var>} = \sqrt{<var>COEFFICIENT * COEFFICIENT</var> \cdot <var>RADICAL</var>}</code></p>
<p><code>\sqrt{<var>NUM</var>} = \sqrt{<var>COEFFICIENT * COEFFICIENT</var>} \cdot \sqrt{<var>RADICAL</var>}</code></p>
<p>Thus, <code>\sqrt{<var>NUM</var>} = <var>COEFFICIENT</var>\sqrt{<var>RADICAL</var>}</code>.</p>
</div>
</div>
</div>
</div>
</body>
</html>
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