<h2>Summary</h2>
<div>
<section>
<ul>
<li>
The number system:
<ul style="list-style-type: square;">
<li>
\(\mathbb{N}\): natural numbers are \(\{1; \; 2; \; 3; \; \ldots\}\)
</li>
<li>
\(\mathbb{N}_0\): whole numbers are \(\{0; \; 1; \; 2; \; 3; \; \ldots\}\)
</li>
<li>
\(\mathbb{Z}\): integers are \(\{\ldots; \; -3; \; -2; \; -1; \; 0; \; 1; \; 2; \; 3; \; \ldots\}\)
</li>
<li>
\(\mathbb{Q}\): rational numbers are numbers which can be written as \(\cfrac{a}{b}\) where \(a\) and \(b\) are integers and \(b\ne 0\), or as a terminating or recurring decimal number.
</li>
<li>
\(\mathbb{Q}'\): irrational numbers are numbers that cannot be written as a fraction with the numerator and denominator as integers. Irrational numbers also include decimal numbers that neither terminate nor recur.
</li>
<li>
\(\mathbb{R}\): real numbers include all rational and irrational numbers.
</li>
<li>
\(\mathbb{R}'\): non-real numbers or imaginary numbers are numbers that are not real.
</li>
</ul>
</li>
<li>
Definitions:
<ul style="list-style-type: square;">
<li>
\({a}^{n}=a\times a\times a\times \cdots \times a (n \text{ times}) (a\in \mathbb{R},n\in \mathbb{N})\)
</li>
<li>
\({a}^{0}=1\) (\(a \ne 0\) because \(0^0\) is undefined)
</li>
<li>
\({a}^{-n}=\cfrac{1}{{a}^{n}}\) (\(a \ne 0\) because \(\dfrac{1}{0}\) is undefined)
</li>
</ul>
</li>
<li>
Laws of exponents:
<ul style="list-style-type: square;">
<li>\({a}^{m} \times {a}^{n}={a}^{m+n}\)</li>
<li>\(\dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n}\)</li>
<li>\({(ab)}^{n}={a}^{n}{b}^{n}\)</li>
<li>\({\left(\cfrac{a}{b}\right)}^{n}=\dfrac{{a}^{n}}{{b}^{n}}\)</li>
<li>\({({a}^{m})}^{n}={a}^{mn}\)</li>
</ul>

where \(a &gt; 0\), \(b &gt; 0\) and \(m, n \in \mathbb{Z}\).</li>
<li>
Rational exponents and surds:
<ul style="list-style-type: square;">
<li>If \(r^n = a\), then \(r = \sqrt[n]{a} \quad (n \geq 2)\)</li>
<li>\(a^{\frac{1}{n}} = \sqrt[n]{a}\)</li>
<li>\(a^{-\cfrac{1}{n}} = (a^{-1})^{\frac{1}{n}} = \sqrt[n]{\dfrac{1}{a}}\)</li>
<li>\(a^{\frac{m}{n}} = (a^{m})^{\frac{1}{n}} = \sqrt[n]{a^m}\)</li>
</ul>

where \(a &gt; 0\), \(r &gt; 0\) and \(m,n \in \mathbb{Z}\), \(n \ne 0\).</li>
<li>
Simplification of surds:
<ul style="list-style-type: square;">
<li>\(\sqrt[n]{a}\sqrt[n]{b} = \sqrt[n]{ab}\)</li>
<li>\(\sqrt[n]{\dfrac{a}{b}} = \dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}\)</li>
<li>\(\sqrt[m]{\sqrt[n]{a}} = \sqrt[mn]{a}\)</li>
</ul>
</li>
</ul>
</section>
</div>

77e14597-4d58-4399-be3f-b1bb5c28bc54

Summary and Main Ideas

Mathematics studies measurement, relationships, and properties of quantities and sets, using numbers and symbols. Arithmetic, algebra, geometry, and calculus are popular topics in mathematics.

Mathematics

Discover how to perform operations with exponents, simplify surds and rationalize denominators, solve surd equations, and apply exponential functions in various scenarios through this tutorial.


Summary and Main Ideas

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