Surds¶
A surd is an irrational number expressed as a root. A simplified surd should have no square factors.
Rules¶
- \(\sqrt{ab} = \sqrt{a}\sqrt{b}\)
- \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\)
Simplifying¶
To simplify a surd, is to extract any rational numbers from it - in practise, removing all square factors.
\[\sqrt{8} = \sqrt{4}\sqrt{2} = 2\sqrt{2}\]
Multiplying¶
To multiply surds, using the following tip will help: Multiply the non-surd first, and the surd second.
\[2\sqrt{3} * 2\sqrt{5} = 4\sqrt{15}\]
Adding¶
To add surds, the \(a + a + a = 3a\) rule needs to be acknowledged.
\[\sqrt{3} + \sqrt{3} = 2\sqrt{3}\]
\[
\begin{align}
\sqrt{2} + \sqrt{8} & = \sqrt{2} + 2\sqrt{2} \\
& = 3\sqrt{2}
\end{align}
\]
Brackets¶
Using brackets and surds is really nothing special.
\[
(\sqrt{2} + 1)(\sqrt{2} - 1)
\]
\[
\begin{array}{c|lc}
& \sqrt{2} & +1 \\
\hline
\sqrt{2} & 2 & +\sqrt{2} \\
-1 & -\sqrt{2} & -1
\end{array}
\]
\[
= 1
\]
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