I understand it a little bit but not that much

<section>
<h2 class="title">Exponent Laws</h2>
<p>There are several laws we can use to make working with exponential numbers easier. Some of these laws might be already be familiar to you, but we list all the laws here for easy reference:</p>
<div>
<ul>
<li>
<p>\({a}^{m}\times {a}^{n}={a}^{m+n}\)</p>
</li>
<li>
<p>\(\dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n}\)</p>
</li>
<li>
<p>\({(ab)}^{n}={a}^{n}{b}^{n}\)</p>
</li>
<li>
<p>\({(\dfrac{a}{b})}^{n}=\dfrac{{a}^{n}}{{b}^{n}}\)</p>
</li>
<li>
<p>\({({a}^{m})}^{n}={a}^{mn}\)</p>
</li>
</ul>
<p>where \(a&gt;0\), \(b&gt;0\) and \(m,n\in \mathbb{R}\)</p>
</div>
<div class="note visit">
<p>The following two videos explain the exponent laws.</p>
<h3>Part 1:</h3>
<iframe allowfullscreen="allowfullscreen" frameborder="0" height="360" src="https://www.youtube.com/embed/kITJ6qH7jS0?rel=0&amp;modestbranding=1&amp;cc_lang_pref=en&amp;cc_load_policy=1" width="640"></iframe>
<h3>Part 2:</h3>
<iframe allowfullscreen="allowfullscreen" frameborder="0" height="360" src="https://www.youtube.com/embed/rEtuPhl6930?rel=0&amp;modestbranding=1&amp;cc_lang_pref=en&amp;cc_load_policy=1" width="640"></iframe></div>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<h3>Question</h3>
<div>
<p>Simplify:</p>
<ol>
<li>
<p>\({2}^{3x}\times {2}^{4x}\)</p>
</li>
<li>
<p>\(\dfrac{4x^{3}}{2x^{5}}\)</p>
</li>
<li>
<p>\(\dfrac{12{p}^{2}{t}^{5}}{3p{t}^{3}}\)</p>
</li>
<li>
<p>\({(3x)}^{2}\)</p>
</li>
<li>
<p>\({({3}^{4}{5}^{2})}^{3}\)</p>
</li>
<li>
<p>\(6p^{0} \times (7p)^{0}\)</p>
</li>
<li>
<p>\({(\dfrac{2xp}{6x^2})}^{3}\)</p>
</li>
<li>
<p>\({({2}^{-2})}^{2x+1}\)</p>
</li>
</ol>
</div>
<ol>
<li>
<p>\({2}^{3x}\times {2}^{4x}={2}^{3x+4x}={2}^{7x}\)</p>
</li>
<li>
<p>\(\dfrac{4x^{3}}{2x^{5}}=2x^{3-5}=2x^{-2}=\dfrac{2}{x^{2}}\)</p>
</li>
<li>
<p>\(\dfrac{12{p}^{2}{t}^{5}}{3p{t}^{3}}=4{p}^{(2-1)}{t}^{(5-3)}=4p{t}^{2}\)</p>
</li>
<li>
<p>\({(3x)}^{2}={3}^{2}{x}^{2}=9{x}^{2}\)</p>
</li>
<li>
<p>\({({3}^{4}\times {5}^{2})}^{3}={3}^{(4\times 3)}\times {5}^{(2\times 3)}={3}^{12}\times {5}^{6}\)</p>
</li>
<li>
<p>\(6p^{0} \times (7p)^{0} = 6(1) \times 1 = 6\)</p>
</li>
<li>
<p>\({(\dfrac{2xp}{6x^2})}^{3}=(\dfrac{p}{3x})^{3}=\dfrac{p^{3}}{27x^{3}}\)</p>
</li>
<li>
<p>\({({2}^{-2})}^{2x+1}=2^{-2(2x+1)}=2^{-4x-2}\)</p>
</li>
</ol>
</div>
</div>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<h3>Question</h3>
<div>
<p>Simplify: \(\dfrac{{2}^{2n}\times {4}^{n}\times 2}{{16}^{n}}\)</p>
</div>
<div>
<h3>Change the bases to prime numbers</h3>
<p>At first glance it appears that we cannot simplify this expression. However, if we reduce the bases to prime bases, then we can apply the exponent laws.</p>

\[\dfrac{{2}^{2n} \times {4}^{n} \times 2}{{16}^{n}} = \dfrac{{2}^{2n} \times {({2}^{2})}^{n} \times {2}^{1}}{{({2}^{4})}^{n}}\]</div>
<div>
<h3>Simplify the exponents</h3>

\begin{align*} &amp; =\dfrac{{2}^{2n} \times {2}^{2n} \times {2}^{1}}{{2}^{4n}} \\ &amp; = \dfrac{{2}^{2n + 2n + 1}}{{2}^{4n}} \\ &amp; = \dfrac{{2}^{4n + 1}}{{2}^{4n}} \\ &amp; ={2}^{4n + 1 - (4n)} \\ &amp; = 2 \end{align*}</div>
</div>
</div>
<div class="ns-school-defn2">
<h4>Tip:</h4>
<p>When you have a fraction that is one term over one term, use the method of Finding Prime Bases - in other words use prime factorisation on the bases.</p>
</div>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<h3>Question</h3>
<div>
<p>Simplify:</p>

\[\dfrac{{5}^{2x - 1} \cdot {9}^{x - 2}}{{15}^{2x - 3}}\]</div>
<div>
<h3>Change the bases to prime numbers</h3>

\begin{align*} \dfrac{{5}^{2x - 1} \cdot {9}^{x - 2}}{{15}^{2x - 3}} &amp; = \dfrac{{5}^{2x - 1} \cdot {({3}^{2})}^{x - 2}}{{(5\times 3)}^{2x - 3}} \\ &amp; = \dfrac{{5}^{2x - 1} \cdot {3}^{2x - 4}}{{5}^{2x - 3} \cdot {3}^{2x - 3}} \end{align*}</div>
<div>
<h3>Subtract the exponents (same base)</h3>

\begin{align*} &amp; = {5}^{(2x - 1) - (2x - 3)} \times {3}^{(2x - 4) - (2x - 3)} \\ &amp; = {5}^{2x - 1 - 2x + 3} \times {3}^{2x - 4 - 2x + 3} \\ &amp; = {5}^{2} \times {3}^{-1} \end{align*}</div>
<div>
<h3>Write the answer as a fraction</h3>

\begin{align*} &amp;= \dfrac{{5}^{2}}{3} \\ &amp; = \dfrac{25}{3} \end{align*}</div>
</div>
</div>
<div class="ns-school-defn2">
<h4>Fact:</h4>
<p>When working with exponents, all the laws of operation for algebra apply.</p>
</div>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<h3>Question</h3>
<div>
<p>Simplify:</p>

\[\dfrac{{2}^{t} - {2}^{t - 2}}{3\cdot {2}^{t} - {2}^{t}}\]</div>
<div>
<h3>Simplify to a form that can be factorised</h3>
<p>For each of the exponent laws we can “undo” the law - in other words we can work backwards. For this expression we can reverse the multiplication law to write \(2^{t - 2}\) as \(2^{t} \cdot 2^{-2}\).</p>

\[\cfrac{{2}^{t} - {2}^{t - 2}}{3\cdot {2}^{t} - {2}^{t}} = \cfrac{{2}^{t} - ({2}^{t}\cdot {2}^{-2})}{3\cdot {2}^{t}-{2}^{t}}\]</div>
<div>
<h3>Take out a common factor</h3>

\[= \cfrac{{2}^{t}(1 - {2}^{-2})}{{2}^{t}(3 - 1)}\]</div>
<div>
<h3>Cancel the common factor and simplify</h3>

\begin{align*} &amp; = \cfrac{1-2^{-2}}{3-1} \\ &amp; = \cfrac{1 - \cfrac{1}{4}}{2}\\ &amp; = \cfrac{3}{4} \times \cfrac{1}{2} \\ &amp; = \cfrac{3}{8} \end{align*}</div>
</div>
</div>
<div class="ns-school-defn2">
<h4>Tip:</h4>
<p>When you have a fraction that has more than one term in the numerator or denominator, change to prime bases if necessary and then factorise.</p>
</div>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<h3>Question</h3>
<div>
<p>Simplify:</p>

\[\dfrac{{9}^{x} - 1}{{3}^{x} + 1}\]</div>
<div>
<h3>Change the bases to prime numbers</h3>

\begin{align*} \dfrac{{9}^{x} - 1}{{3}^{x} + 1} &amp; = \dfrac{{({3}^{2})}^{x} - 1}{{3}^{x} + 1} \\ &amp; = \dfrac{{3}^{2x} - 1}{{3}^{x} + 1} \qquad \text{Recognise that } 3^{2x} = (3^{x})^{2} \end{align*}</div>
<div>
<h3>Factorise using the difference of squares</h3>

\[= \dfrac{({3}^{x} - 1)({3}^{x} + 1)}{{3}^{x} + 1}\]</div>
<h3>Cancel the common factor and simplify</h3>
<p>\[= {3}^{x}-1\]</p>
</div>
</div>
</section>

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Exponent Laws

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Exponent Laws

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