Section 3.3 Section 3.3 Roots and Radicals
Subsection 1. Use the definition of root
Because \(~(\sqrt{a})(\sqrt{a})=a,~\) it is also true that \(~\dfrac{a}{\sqrt{a}}= \sqrt{a}\text{.}\)
Subsubsection Examples
Example 3.29.
Simplify. Do not use a calculator!
- \(\displaystyle \left(\sqrt{7}\right)\left(\sqrt{7}\right)\)
- \(\displaystyle \sqrt{n}(\sqrt{n})\)
By the definition of square root, \(\sqrt{a}\) is a number whose square is \(a\text{.}\)
- \(\displaystyle \left(\sqrt{7}\right)\left(\sqrt{7}\right)=7\)
- \(\displaystyle \sqrt{n}(\sqrt{n})=n\)
Example 3.30.
Simplify. Do not use a calculator!
- \(\displaystyle \left(\sqrt[3]{5}\right)^3\)
- \(\displaystyle \left(\sqrt[3]{4}\right)\left(\sqrt[3]{4}\right)\left(\sqrt[3]{4}\right)\)
By the definition of cube root, \(\sqrt[3]{a}\) is a number whose cube is \(a\text{.}\)
- \(\displaystyle \left(\sqrt[3]{5}\right)^3=5\)
- \(\displaystyle \left(\sqrt[3]{4}\right)\left(\sqrt[3]{4}\right)\left(\sqrt[3]{4}\right)=4\)
Example 3.31.
Simplify. Do not use a calculator!
- \(\displaystyle \dfrac{3}{\sqrt{3}}\)
- \(\displaystyle \dfrac{p}{\sqrt{p}}\)
- \(\displaystyle \dfrac{3}{\sqrt{3}} = \dfrac{\sqrt{3}\sqrt{3}}{\sqrt{3}} = \sqrt{3}\)
- \(\displaystyle \dfrac{p}{\sqrt{p}} = \dfrac{\sqrt{p}\sqrt{p}}{\sqrt{p}} = \sqrt{p}\)
Subsubsection Exercises
Checkpoint 3.32.
Simplify. Do not use a calculator!
- \(\displaystyle \sqrt{5}(\sqrt{5})\)
- \(\displaystyle \sqrt{x}(\sqrt{x})\)
- \(\displaystyle 5\)
- \(\displaystyle x\)
Checkpoint 3.33.
Simplify. Do not use a calculator!
- \(\displaystyle \left(\sqrt[3]{9}\right)\left(\sqrt[3]{9}\right)\left(\sqrt[3]{9}\right)\)
- \(\displaystyle \left(\sqrt[3]{20}\right)^3\)
- \(\displaystyle 9\)
- \(\displaystyle 20\)
Checkpoint 3.34.
Simplify. Do not use a calculator!
- \(\displaystyle \dfrac{10}{\sqrt{10}}\)
- \(\displaystyle \dfrac{H}{\sqrt{H}}\)
- \(\displaystyle \sqrt{10}\)
- \(\displaystyle \sqrt{H}\)
Subsection 2. Approximate rational numbers
Rational numbers are the integers and common fractions; we can represent them precisely in decimal form. But the best we can do for an irrational number is to write an approximate decimal form by rounding.
Subsubsection Examples
Example 3.35.
Identify each number as rational or irrational.
- \(\displaystyle \sqrt{6}\)
- \(\displaystyle \dfrac{-5}{3}\)
- \(\displaystyle \sqrt{16}\)
- \(\displaystyle \sqrt{\dfrac{5}{9}}\)
- Irrational: \(~\sqrt{6}~\) is not the quotient of two integers.
Rational: \(~\dfrac{-5}{3}~\) is the quotient of two integers.
- Rational: \(~\sqrt{16} = 4~\) is an integer.
Irrational: \(~\sqrt{\dfrac{5}{9}} = \dfrac{\sqrt{5}}{3}~\text{,}\) but \(\sqrt{5}\) is irrational.
Example 3.36.
Give a decimal approximation rounded to thousandths.
- \(\displaystyle 5\sqrt{3}\)
- \(\displaystyle \dfrac{-2}{3}\sqrt{21}\)
- \(\displaystyle 2+\sqrt[3]{5}\)
Use a calculator to evaluate.
- Enter \(~5~ \boxed{\sqrt{}}~3\) ENTER and round to three decimal places: \(~8.660 \)
- Enter (-) \(2\) \(\boxed{\sqrt{}}~\) \(21\) ) ÷ \(3\) ENTER and round to three decimal places: \(~-3.055\)
- Enter \(2\) + MATH 4 \(5\) ENTER and round to three decimal places: \(~3.710\)
Subsubsection Exercises
Checkpoint 3.37.
Identify each number as rational or irrational.
- \(\displaystyle \sqrt{250}\)
- \(\displaystyle \dfrac{\sqrt{3}}{2}\)
- \(\displaystyle \dfrac{\sqrt{81}}{4}\)
- \(\displaystyle \sqrt[3]{16}\)
- Irrational
- Irrational
- Rational
- Irrational
Checkpoint 3.38.
Give a decimal approximation rounded to thousandths.
- \(\displaystyle -6\sqrt[3]{5}\)
- \(\displaystyle \dfrac{3}{5}\sqrt{76}\)
- \(\displaystyle 7-\sqrt{19}\)
- \(\displaystyle -10.260\)
- \(\displaystyle 5.231\)
- \(\displaystyle 2.641\)
Subsection 3. Use the order of operations
In the order of operations, simplifying radicals comes after what's inside parentheses (or fraction bars) and before products and quotients.
Subsubsection Examples
Example 3.39.
Simplify each expression. Do not use a calculator!
- \(\displaystyle \dfrac{4-\sqrt{64}}{2}\)
- \(\displaystyle -2\left(3\sqrt{16}-\sqrt{3(27)}\right)\)
- \(\displaystyle 6-3\sqrt[3]{27-7(5)}\)
-
We start by simplifying the numerator.
\begin{align*} \dfrac{4-\sqrt{64}}{2} \amp = \dfrac{4-8}{2} \amp\amp \blert{\text{Evaluate the radical, then subtract.}}\\ \amp = \dfrac{-4}{2}=-2 \amp\amp \blert{\text{Reduce the fraction.}} \end{align*} -
We start by simplifyng what's inside parentheses.
\begin{align*} -2\amp\left(3\sqrt{16} -\sqrt{3(27)}\right) \amp\amp \blert{\text{Evaluate the radicals.}}\\ \amp = -2(3 \cdot 4-\sqrt{81}) \amp\amp \blert{\text{Simplify inside the parentheses.}}\\ \amp = -2(12-9)=-6 \end{align*} -
We start by simplifying the radicand.
\begin{align*} 6-3\sqrt[3]{27-7(5)} \amp = 6-3\sqrt[3]{27-35} \amp\amp \blert{\text{Subtract under the radical.}}\\ \amp = 6-3\sqrt[3]{-8} \amp\amp \blert{\text{Evaluate the radical.}}\\ \amp =6-3(-2)=12 \end{align*}
Example 3.40.
Simplify each expression. Round your answer to hundredths.
- \(\displaystyle \dfrac{8-2\sqrt{2}}{4}\)
- \(\displaystyle 2+6\sqrt[3]{-25}\)
-
Do not start with "\(8-2\)"! Evaluate \(\sqrt{2}\) first, then multiply by 2, and subtract the result from 8. Once the numerator is simplified, divide by 4.
On a calculator, enter
\(\qquad\qquad\) ( \(8\) - \(2~\boxed{\sqrt{}}\) ) ) ÷ \(4\) ENTER
and round to two decimal places: \(~1.29\)
-
Evaluate the cube root, multiply by 6, then add the result to 2.
On a calculator, enter
\(\qquad\qquad 2\) + \(6\) MATH 4 - \(25\) ENTER
and round to two decimal places: \(~{-15.54}\)
Subsubsection Exercise
Checkpoint 3.41.
Simplify each expression. Do not use a calculator!
- \(\displaystyle \dfrac{36}{6+\sqrt{36}}\)
- \(\displaystyle 10+2(3-\sqrt{169})\)
- \(\displaystyle \dfrac{3+\sqrt[3]{-729}}{6-\sqrt[3]{-27}}\)
- \(\displaystyle 3\)
- \(\displaystyle -10\)
- \(\displaystyle \dfrac{-2}{3}\)
Checkpoint 3.42.
Simplify each expression. Round your answer to hundredths.
- \(\displaystyle \dfrac{6+9\sqrt{3}}{3}\)
- \(\displaystyle -1-3\sqrt[3]{120}\)
- \(\displaystyle 7.20\)
- \(\displaystyle -15.80\)