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<!DOCTYPE html>
<html data-require="math">
<head>
<meta charset="UTF-8" />
<title>Imaginary unit powers</title>
<script src="../khan-exercise.js"></script>
</head>
<body>
<div class="exercise">
<div class="problems">
<div id="simple-i-power-less-than-4" data-weight="2">
<div class="vars">
<var id="EXP">randRange( 1, 4 )</var>
<var id="SOLUTION">(function() {
switch ( EXP % 4 ) {
case 0: return '1';
case 1: return 'i';
case 2: return '-1';
case 3: return '-i';
}
})()</var>
</div>
<div class="problem">
<p class="question">
Simplify.
</p>
<p>
<code>i ^ {<var>EXP</var>}</code>
</p>
</div>
<div class="solution"><code><var>SOLUTION</var></code></div>
<ul class="choices" data-category="true">
<li><code>1</code></li>
<li><code>i</code></li>
<li><code>-1</code></li>
<li><code>-i</code></li>
</ul>
<div class="hints">
<p data-if="EXP === 1">
Anything to the first power is the number itself.
</p>
<p data-if="EXP >= 2">
The most important property of the imaginary unit <code>i</code> is
that <code>\color{<var>BLUE</var>}{i ^ 2} = \color{<var>ORANGE</var>}{-1}</code>.
</p>
<p data-if="EXP === 3">
<code>i ^ 3 = (\color{<var>ORANGE</var>}{i ^ 2}) \cdot i = (\color{<var>BLUE</var>}{-1}) \cdot i = -i</code>
</p>
<p data-if="EXP === 4">
<code>i ^ 4 = (\color{<var>ORANGE</var>}{i ^ 2}) ^ 2 = (\color{<var>BLUE</var>}{-1}) ^ 2 = 1</code>
</p>
<p>
<code>i ^ <var>EXP</var> = <var>SOLUTION</var></code>
</p>
</div>
</div>
<div id="simple-i-power-more-than-4" data-weight="5">
<div class="vars">
<var id="EXP">4 + randRange( 1, 30 )</var>
<var id="SOLUTION">(function() {
switch ( EXP % 4 ) {
case 0: return '1';
case 1: return 'i';
case 2: return '-1';
case 3: return '-i';
}
})()</var>
</div>
<div class="problem">
<p class="question">
Simplify.
</p>
<p>
<code>i ^ {<var>EXP</var>}</code>
</p>
</div>
<div class="solution"><code><var>SOLUTION</var></code></div>
<ul class="choices" data-category="true">
<li><code>1</code></li>
<li><code>i</code></li>
<li><code>-1</code></li>
<li><code>-i</code></li>
</ul>
<div class="hints">
<p>
The most important property of the imaginary unit <code>i</code> is
that <code>\color{<var>BLUE</var>}{i ^ 2} = \color{<var>ORANGE</var>}{-1}</code>.
</p>
<p>
When this property is plugged into <code>i ^ 4</code>, we get:
<code>i ^ 4 = (\color{<var>BLUE</var>}{i ^ 2}) ^ 2 = (\color{<var>ORANGE</var>}{-1}) ^ 2 = 1</code>
</p>
<p>
So, we can reduce the exponent by multiples of 4 and have the same result.
</p>
<p>
The remainder after dividing <var>EXP</var> by 4 is <var>EXP % 4</var>, <!-- maybe add a reference to mod here... probably after a mod module is made -->
so <code>i ^ {<var>EXP</var>} = i ^ {<var>EXP % 4</var>}</code>.
</p>
<div data-if="(EXP % 4) === 0">
<p>
Any number but zero to the zeroth power is one.
</p>
<p>
<code>i ^ 0 = 1</code>
</p>
</div>
<div data-if="(EXP % 4) === 1">
<p>
Anything to the first power is the number itself.
</p>
<p>
<code>i ^ 1 = i</code>
</p>
</div>
<p data-if="(EXP % 4) >= 2">
As stated above, <code>\color{<var>BLUE</var>}{i ^ 2} = \color{<var>ORANGE</var>}{-1}</code>.
</p>
<p data-if="(EXP % 4) === 3">
<code>i ^ 3 = (\color{<var>BLUE</var>}{i ^ 2}) \cdot i = (\color{<var>ORANGE</var>}{-1}) \cdot i = -i</code>
</p>
<p>
<code>i ^ {<var>EXP</var>} = i ^ {<var>EXP % 4</var>} = <var>SOLUTION</var></code>.
</p>
</div>
</div>
</div>
</div>
</body>
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