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\title{Detailed Explanation of Adding and Subtracting Surds}
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\section*{Understanding the Concept}
Surds are irrational numbers expressed in their root form, such as \( \sqrt{2} \) or \( \sqrt{5} \), which cannot be simplified into exact decimals. When adding or subtracting surds, specific rules must be followed to ensure correct simplification.

\section*{1. The Importance of Like Terms}
Just as in algebra, you can only add or subtract surds if they are \textbf{like terms}—meaning they have the same number under the square root sign.
\begin{itemize}
\item Example 1: \( 2\sqrt{5} + 7\sqrt{5} = 9\sqrt{5} \), since both terms contain \( \sqrt{5} \).
\item Example 2: \( 2\sqrt{5} + 7\sqrt{6} \) cannot be simplified further, as \( \sqrt{5} \) and \( \sqrt{6} \) are different surds.
\end{itemize}

\section*{2. The Need for Simplification}
In many cases, surds must be simplified before they can be added or subtracted. This involves breaking them down into factors, identifying perfect square factors, and simplifying accordingly.

\subsection*{How to Simplify Surds Before Addition or Subtraction}
\begin{enumerate}
\item \textbf{Identify perfect square factors}: Find square numbers within the radicand (the number inside the square root).
\item \textbf{Rewrite the surd}: Express the surd in terms of the square root of its factors.
\item \textbf{Extract the square root of perfect squares}: Factor out and simplify.
\item \textbf{Combine like terms}: Add or subtract the surds if they are now similar.
\item \textbf{Leave unlike terms separate}: If surds remain different, they cannot be combined.
\end{enumerate}

\section*{Examples of Adding and Subtracting Surds}

\subsection*{Example 1: Simplify \( 10\sqrt{2} – 7\sqrt{2} \)}
Since both terms contain \( \sqrt{2} \), simply subtract the coefficients:
\[
10\sqrt{2} – 7\sqrt{2} = 3\sqrt{2}
\]

\subsection*{Example 2: Simplify \( 6\sqrt{24} \)}
\[
\sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6}
\]
Now multiply by 6:
\[
6 \times 2\sqrt{6} = 12\sqrt{6}
\]

\subsection*{Example 3: Simplify \( \sqrt{27} – \sqrt{12} \)}
\[
\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}
\]
\[
\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}
\]
Subtracting:
\[
3\sqrt{3} – 2\sqrt{3} = \sqrt{3}
\]

\subsection*{Example 4: Simplify \( \sqrt{63} + \sqrt{28} – \sqrt{175} \)}
\[
\sqrt{63} = \sqrt{9 \times 7} = 3\sqrt{7}
\]
\[
\sqrt{28} = \sqrt{4 \times 7} = 2\sqrt{7}
\]
\[
\sqrt{175} = \sqrt{25 \times 7} = 5\sqrt{7}
\]
Now, add and subtract:
\[
3\sqrt{7} + 2\sqrt{7} – 5\sqrt{7} = 0
\]

\subsection*{Example 5: Simplify \( \sqrt{20} + \sqrt{45} + \sqrt{48} \)}
\[
\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}
\]
\[
\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}
\]
\[
\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}
\]
Combining like terms:
\[
2\sqrt{5} + 3\sqrt{5} + 4\sqrt{3} = 5\sqrt{5} + 4\sqrt{3}
\]
Since \( \sqrt{5} \) and \( \sqrt{3} \) are not like terms, they remain separate.

\section*{3. Rules to Remember When Working with Surds}
\begin{itemize}
\item \textbf{Only like surds can be added or subtracted}.
\item \textbf{Always simplify surds first} if possible.
\item \textbf{Break down the radicand into perfect squares} to simplify further.
\item \textbf{Leave unlike surds as separate terms}.
\item \textbf{Follow basic surd rules}, such as \( a\sqrt{b} = \sqrt{a^2b} \), which helps in breaking down complex expressions.
\end{itemize}

\section*{Summary of Steps for Adding and Subtracting Surds}
\begin{enumerate}
\item \textbf{Check if the surds have the same radicand}.
\item \textbf{Simplify each surd individually} by breaking them into their prime factors.
\item \textbf{Extract perfect square factors} and simplify.
\item \textbf{Combine like surds by adding or subtracting coefficients}.
\item \textbf{Leave the final answer in its simplest form}.
\end{enumerate}

Mastering these techniques allows for efficient manipulation of surds in mathematical problems, including algebra, geometry, and higher-level mathematics.

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