Unit 6.1 Introduction to Radicals and Simplifying Radical Expressions
Radicals
In mathematics, a radical expression is an expression that includes a radical symbol (√), which indicates a root. The most common radical is the square root, but radicals can also denote cube roots, fourth roots, and higher-order roots. This is often called the “nth” root
Components of a Radical Expression
- Radical Symbol (√): Indicates the root being taken.
- Radicand: The number or expression inside the radical symbol.
- Index: Indicates the degree of the root.
- If no index is shown, it is assumed to be 2 (square root)
Radical Rules
Simplifying Radical Expressions with Perfect nth Roots
[latex]\sqrt[n]{a^n} = a[/latex]
Product Rule of Radical Expressions
[latex]\sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b}[/latex]
Quotient Rule of Radical Expressions
[latex]\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}, \quad b \neq 0[/latex]
[latex]\sqrt{16} = 4[/latex]
[latex]\sqrt[3]{27} = 3[/latex]
[latex]\sqrt[4]{16} = 2[/latex]
Break apart the larger number into smaller components, find a value that is a perfect root
For example 72 can be broken down into 36 times 2. 36 is a perfect square so we are able to simplify it
[latex]\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}[/latex]