4 Radicals and Rational Exponents
Learning Objectives
In this section students will:
- 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.
A hardware store sells 16-ft ladders and 24-ft ladders. A window is located 12 feet above the ground. A ladder needs to be purchased that will reach the window from a point on the ground 5 feet from the building. To find out the length of ladder needed, we can draw a right triangle as shown in (Figure), and use the Pythagorean Theorem.
Evaluating Square Roots
When the square root of a number is squared, the result is the original number. Since[latex]\,{4}^{2}=16,[/latex]the square root of[latex]\,16\,[/latex]is[latex]\,4.\,[/latex]The square root function is the inverse of the squaring function just as subtraction is the inverse of addition. To undo squaring, we take the square root.
In general terms, if[latex]\,a\,[/latex]is a positive real number, then the square root of[latex]\,a\,[/latex]is a number that, when multiplied by itself, gives[latex]\,a.\,[/latex]The square root could be positive or negative because multiplying two negative numbers gives a positive number. The principal square root is the nonnegative number that when multiplied by itself equals[latex]\,a.\,[/latex]The square root obtained using a calculator is the principal square root.
The principal square root of[latex]\,a\,[/latex]is written as[latex]\,\sqrt{a}.\,[/latex]The symbol is called a radical, the term under the symbol is called the radicand, and the entire expression is called a radical expression.
Principal Square Root
The principal square root of[latex]\,a\,[/latex]is the nonnegative number that, when multiplied by itself, equals[latex]\,a.\,[/latex]It is written as a radical expression, with a symbol called a radical over the term called the radicand:[latex]\,\sqrt{a}.[/latex]
Does[latex]\,\sqrt{25}=±5?[/latex]
No. Although both[latex]\,{5}^{2}\,[/latex]and[latex]\,{\left(-5\right)}^{2}\,[/latex]are[latex]\,25,[/latex]the radical symbol implies only a nonnegative root, the principal square root. The principal square root of 25 is[latex]\,\sqrt{25}=5.[/latex]
Evaluating Square Roots
Evaluate each expression.
- [latex]\sqrt{100}[/latex]
- [latex]\sqrt{\sqrt{16}}[/latex]
- [latex]\sqrt{25+144}[/latex]
- [latex]\sqrt{49}-\sqrt{81}[/latex]
[hidden-answer a=”fs-id1441682″]
- [latex]\sqrt{100}=10\,[/latex]because[latex]\,{10}^{2}=100[/latex]
- [latex]\sqrt{\sqrt{16}}=\sqrt{4}=2\,[/latex]because[latex]\,{4}^{2}=16\,[/latex]and[latex]\,{2}^{2}=4[/latex]
- [latex]\sqrt{25+144}=\sqrt{169}=13\,[/latex]because[latex]\,{13}^{2}=169[/latex]
- [latex]\sqrt{49}-\sqrt{81}=7-9=-2\,[/latex]because[latex]\,{7}^{2}=49\,[/latex]and[latex]\,{9}^{2}=81[/latex]
[/hidden-answer]
For[latex]\,\sqrt{25+144},[/latex]can we find the square roots before adding?
No.[latex]\,\sqrt{25}+\sqrt{144}=5+12=17.\,[/latex]