\begin{definition}[Radical] An expression of the form \$\sqrt[n]{a}\$, where \$n\$ is the \textbf{index} and \$a\$ is the \textbf{radicand}. \end{definition}
\begin{note} If the \textbf{index} is not specified, it is assumed to be \$2\$. \end{note}
Simplifying Radicals
To simplify a radical:
- \textbf{Factor} the radicand into its \textbf{prime factors}.
- \textbf{Group} the factors into \textbf{pairs} (or groups of \$n\$ for \$\sqrt[n]{a}\$).
- \textbf{Move} each group \textbf{outside} the radical, leaving \textbf{unpaired} factors \textbf{inside}.
\begin{example} Simplify \$\sqrt{72}\$.
- Factor \$72\$ into primes: \$72 = 2^3 \cdot 3^2\$.
- Group the factors: \$72 = (2^2) \cdot 2 \cdot (3^2)\$.
- Move pairs outside: \$\sqrt{72} = 2 \cdot 3 \cdot \sqrt{2} = 6\sqrt{2}\$. \end{example}
\begin{note} The \textbf{simplified form} of a radical has no \textbf{perfect powers} of the index inside the radical. \end{note}
Operations with Radicals
Multiplying and Dividing Radicals
\begin{note} Radicals can be multiplied or divided if they have the \textbf{same index}. \end{note}
- \textbf{Multiply} or \textbf{divide} the radicands.
- Simplify the result.
\begin{example} Multiply \$\sqrt{3} \cdot \sqrt{12}\$.
- Multiply the radicands: \$\sqrt{3 \cdot 12} = \sqrt{36}\$.
- Simplify: \$\sqrt{36} = 6\$. \end{example}
\begin{warning} Ensure the \textbf{indices} are the same before performing operations. \end{warning}
Adding and Subtracting Radicals
\begin{note} Radicals can be added or subtracted only if they have the \textbf{same index} and \textbf{same radicand}. \end{note}
- Combine the \textbf{coefficients} of like radicals.
- Keep the radical part unchanged.
\begin{example} Add \$3\sqrt{5} + 2\sqrt{5}\$.
- Combine coefficients: \$(3 + 2)\sqrt{5} = 5\sqrt{5}\$. \end{example}
\begin{warning} Do not combine radicals with \textbf{different radicands}. \end{warning}
Rationalizing the Denominator
\begin{definition}[Rationalizing the Denominator] The process of eliminating radicals from the denominator of a fraction. \end{definition}
- Multiply the numerator and denominator by a \textbf{conjugate} or appropriate radical.
- Simplify the expression.
\begin{example} Rationalize \$\frac{1}{\sqrt{2}}\$.
- Multiply by \$\frac{\sqrt{2}}{\sqrt{2}}\$: \$\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}\$. \end{example}
\begin{warning} Ensure the expression remains \textbf{equivalent} after rationalizing. \end{warning}
\begin{self_review}
- Simplify \$\sqrt{50}\$.
- Multiply \$\sqrt{2} \cdot \sqrt{8}\$.
- Add \$3\sqrt{7} + 5\sqrt{7}\$.
- Rationalize \$\frac{1}{\sqrt{3}}\$. \end{self_review}
\begin{tok} How do different cultures and mathematical traditions approach the concept of radicals and irrational numbers? \end{tok}
