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Section 5.3 Some Basic Functions

Subsection 1. Evaluate cube roots

Subsubsection Examples

It is a good idea to become familiar with the first few perfect cubes:

\begin{equation*} 2^3=8~~~~~~3^3=27~~~~~~4^3=64~~~~~~5^3=125~~~~~~6^3=216 \end{equation*}

and so on.

Example 5.12.

Evaluate each cube root.

  1. \(\displaystyle \sqrt[3]{64}\)
  2. \(\displaystyle \sqrt[3]{-125}\)
  3. \(\displaystyle \sqrt[3]{1}\)
  4. \(\displaystyle \sqrt[3]{\dfrac{-1}{8}}\)
Solution
  1. \(4^3=64,~\) so \(~\sqrt[3]{64}=4\)

  2. The cube root of a negative number is negative.

    \begin{equation*} (-5)^3=-125,~~\text{so}~~\sqrt[3]{-125}=-5 \end{equation*}
  3. \(1^3=1,~\) so \(~\sqrt[3]{1}=1\)

  4. We can take the cube root of a fraction by taking the cube root of its numerator and denominator.

    \begin{equation*} \sqrt[3]{\dfrac{-1}{8}} = \dfrac{\sqrt[3]{-1}}{\sqrt[3]{8}} = \dfrac{-1}{2} \end{equation*}
Example 5.13.

Use a calculator to evaluate the cube root. Round to thousandths.

  1. \(\displaystyle \sqrt[3]{347}\)
  2. \(\displaystyle \sqrt[3]{0.85}\)
  3. \(\displaystyle \sqrt[3]{-9}\)
Solution

On a scientific calculator, look for the key labeled \(\boxed{\sqrt[3]{\hphantom{1}\vphantom{1}}}\text{.}\) On a graphing calculator, press MATH 4

  1. \(\displaystyle \sqrt[3]{347} \approx 7.027\)
  2. \(\displaystyle \sqrt[3]{0.85} \approx 0.947\)
  3. \(\displaystyle \sqrt[3]{-9} \approx -2.080\)

Subsubsection Exercises

Evaluate \(\sqrt[3]{-0.5}\text{.}\) Round to thousandths.

Answer
\(-0.794\)

Evaluate \(\sqrt[3]{81}\text{.}\) Round to thousandths.

Answer
\(4.327\)

Subsection 2. Evaluate absolute values

Subsubsection Examples

The definition of how to take an absolute value may look complicated, but it just says two things:

  1. If the number is positive, leave it alone.
  2. If the number is negative, put another negative in front, which will make the number positive.
Example 5.16.

Simplify each expression.

  1. \(\displaystyle |-3|\)
  2. \(\displaystyle -|3|\)
  3. \(\displaystyle -(-3)\)
  4. \(\displaystyle -|-3|\)
Solution

The absolute value of any number is positive (or zero). We can think of the absolute value of a number as its distnce from on a number line.

  1. \(-3\) is 3 units from 0, so \(|-3|=3\text{.}\)
  2. \(-|3|\) is the opposite of \(|3|\text{,}\) so \(-|3|=-3\text{.}\)
  3. The opposite of \(-3\) is \(3\text{,}\) so \(-(-3)=3\text{.}\)
  4. \(-|-3|\) is the opposite of \(|-3|\text{,}\) so \(-|-3|=-3\text{.}\)
Example 5.17.

Suppose \(x\) represents \(-8\text{.}\) Evaluate each expression.

  1. \(\displaystyle -x\)
  2. \(\displaystyle |x|\)
  3. \(\displaystyle |-x|\)
Solution
  1. \(\displaystyle -x = -(-8) = 8\)
  2. \(\displaystyle |x| = |-8| = 8\)
  3. \(\displaystyle |-x| = |-(-8)| = 8\)

Subsubsection Exercises

Simplify \(-|-12|\text{.}\)

Answer
\(-12\)

Simplify \(|-25|\text{.}\)

Answer
\(25\)

Simplify \(-(-90)|\text{.}\)

Answer
\(90\)

Subsection 3. Use the order of operations in evaluation

Recall the order of operations:

  1. Simplify what's inside parentheses (or absolute value bars) first.
  2. Next evaluate all powers and roots.
  3. Then perform all multiplications and divisions in order from left to right.
  4. Finally, perform all additionas and subtractions in order from left to right.

Subsubsection Examples

Example 5.21.

Simplify \(~|2|-4|3-8|\)

Solution

Absolute value bars are a grouping device. We simplify expressions within absolute value bars first.

\begin{align*} |2|-4|\alert{3-8}| \amp = |2|- 4|\alert{-5}| \amp \amp \blert{\text{Evaluate absolute values.}}\\ \amp = 2-4(5) \amp \amp \blert{\text{Multiply.}}\\ \amp =2-20=18 \end{align*}
Example 5.22.

Simplify \(~\dfrac{8-2\sqrt[3]{11.375+2.5^3}}{8-4}\)

Solution

Simplify the expression under the radical first.

\begin{align*} \dfrac{8-2\sqrt[3]{\alert{11.375+2.5^3}}}{8-4} \amp = \dfrac{8-2\sqrt[3]{\alert{27}}}{8-4} \amp \amp ~\blert{\text{Evaluate the radical.}}\\ amp =\dfrac{8-2(3)}{8-4} \amp \amp \begin{array}{l} \blert{\text{Simplify numerator }}\\ \blert{\text{and denominator.}} \end{array}\\ \amp =\dfrac{8-6}{4}\\ \amp = \dfrac{2}{4} = \dfrac{1}{2} \end{align*}

Subsubsection Exercises

Simplify \(~~3\sqrt[3]{\dfrac{125}{216}} + \dfrac{4}{5} \sqrt[3]{-512}~~~\text{.}\) Follow the order of operations.

Answer

\(\dfrac{-39}{10}\)

Simplify \(~~-3|3-6|-4|-4-3|~~~\text{.}\) Follow the order of operations.

Answer

\(-37\)