1.3 Radical and Rational Exponents
Learning Objectives
- Evaluate square roots.
- Use the product rule to simplify square roots.
- Use the quotient rule to simplify square roots.
- Add and subtract square roots.
- Rationalize denominators.
- Use rational roots.
Terminology
Radical expression is an expression containing a radical sign,
[latex]\sqrt{4}[/latex]
Radicand is the expression under a radical sign
4 is the radicand for the expression [latex]\sqrt{4}[/latex]
Note that if the radicand of a square root is a negative number, the radical is not a real number.
Square Root of a Number
If [latex]b^2 = a[/latex], then b is a square root of a.
| Meaning | Principal Root Positive Square Root | Secondary Root Negative Square Root | The positive and negative square roots |
|---|---|---|---|
| Symbol | [latex]\sqrt{x}[/latex] | [latex]-\sqrt{x}[/latex] | [latex]\pm\sqrt{x}[/latex] |
| Example | [latex]\sqrt{4}[/latex] | [latex]-\sqrt{4}[/latex] | [latex]\pm\sqrt{4}[/latex] |
Square roots of perfect square radicands simplify to rational numbers
Square roots of numbers that are not perfect squares are irrational numbers.
Using a calculator to simplify gives an approximation for these irrational numbers. Leaving it in a simplified radical form is called an exact solution.
Product Property for Square Roots
[latex]\Large\sqrt{a \cdot b} \: = \: \sqrt{a} \cdot \sqrt{b}[/latex]
Example 1.3.1
Use the Product Property to simplify the following roots:
a. [latex]\sqrt{36}=\sqrt{4 \cdot 9}[/latex]
b. [latex]\sqrt{100}=\sqrt{4 \cdot 25}[/latex]
Explanation:
a. While [latex]\sqrt{36}=6[/latex] is a perfect square, we can also use to demonstrate the product property.
[latex]\large\sqrt{36}=\sqrt{4 \cdot 9}[/latex]
[latex]\large\sqrt{36}=\sqrt{4} \cdot \sqrt{9}[/latex]
[latex]\large 6=2 \cdot 3[/latex]
Quotient Rule for Square Roots
[latex]\Large\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}[/latex]
Example 1.3.2
Use the Quotient Rule to simplify [latex]\sqrt{\frac{7}{36}}[/latex]
The quotient rule allows us to rewrite the expression as a quotient of two square root expressions.
[latex]\large\frac{\sqrt7}{\sqrt{36}}[/latex]
Now, since 36 is a perfect square we can rewrite the expression again.
[latex]\large\frac{\sqrt7}{6}[/latex]
Example 1.3.3
Use the Quotient Rule to Simplify
Cube Root of a Number (move to after radical operation with nth root)
[latex]\large\sqrt[3]{a}=b \text{ if and only if } b^3 = a[/latex]
