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Section 4.1 Exponential Growth and Decay

Subsection 1. Compute percent increase and decrease

To calculate an increase of \(r\)%, we write the percent as a decimal and multiply the old amount by \(1+r\text{.}\) To calculate a decrease we multiply the old amount by \(1-r\text{.}\)

Subsubsection Examples

Example 4.1.

A loaf of bread cost $3.00 last month, but this year the price rose by 6%. What should you multiply by to find the new price? What is the new price?

Solution

To get the new price, we multiply by 1.06 to get

\begin{equation*} 1.06(3.00)=3.18 \end{equation*}

The new price is $3.18.

Example 4.2.

Priceco is offering a 15% discount off the regular price of $180 for a ceiling fan. What should you multiply by to find the new price? What is the new price?

Solution

To get the new price, we multiply by \(1-0.15\text{,}\) or 0.85, to get

\begin{equation*} 0.85(180)=153 \end{equation*}

so the new price is $153.

Subsubsection Exercises

Muriel's rent was increased by 8% from $650 per month. What should you multiply by to find her new rent? What is her new rent?

Answer

\(1.08\text{,}\) $702

A brand new SUV loses 18% of its value as soon as you drive it off the lot. If your SUV cost $35,000, what should you multiply to find its new value? What is its new value?

Answer

\(0.82\text{,}\) $28,700

Subsection 2. Use the order of operations

Recall that evaluating powers comes before multiplication in the order of operations.

Subsubsection Examples

Example 4.5.

Simplify.

  1. \(\displaystyle -4-2^3\)
  2. \(\displaystyle -4(-2)^3\)
  3. \(\displaystyle (-4-2)^3\)
Solution
  1. Compute \(2^3\) first, then subtract the result from \(-4\text{:}\)
    \begin{equation*} -4-2^3=-4-8=-12 \end{equation*}
  2. Compute \((-2)^3\) first, then multiply the result by \(-4\text{:}\)
    \begin{equation*} -4(-2)^3=-4(-8)=32 \end{equation*}
  3. Compute \((-4-2)\) first, then cube the result:
    \begin{equation*} (-4-2)^3=(-6)^3=-216 \end{equation*}
Example 4.6.

Evaluate for\(x=6\text{.}\) Round your answers to hundredths.

  1. \(\displaystyle 12(1.05)^x\)
  2. \(\displaystyle 12(1+x/100)^5\)
Solution

  1. Follow the order of operations. Compute the power first:

    \begin{equation*} 12(1.05)^{\alert{6}} = 12(1.3400956...) = 16.08 \end{equation*}

  2. Follow the order of operations. Compute the power first:

    \begin{equation*} 12(1+\alert{6}/100)^5 = 12(1.06)^5 = 12(1.3382255...) = 16.06 \end{equation*}

Subsubsection Exercises

Simplify. Round your answers to the nearest whole number.

  1. \(\displaystyle 450(1-0.12)^4\)
  2. \(\displaystyle 180-80(1+0.25)^3\)
Answer
  1. \(\displaystyle 270\)
  2. \(\displaystyle 24\)

Evaluate for \(x=-3,~y=-2\text{.}\)

  1. \(\displaystyle -2x^2+y^3\)
  2. \(\displaystyle 4(x-y)(x+2y)\)
Answer
  1. \(\displaystyle -26\)
  2. \(\displaystyle 28\)

Subsection 3. Raise fractions to powers

Subsubsection Examples

Example 4.9.

Complete the table of powers. As the exponent increases, do the powers increase or decrease?

  1. \(~~x~~\) \(~~1~~\) \(~~2~~\) \(~~3~~\) \(~~4~~\)
    \(\left(\dfrac{2}{3}\right)^x\) \(\) \(\) \(\) \(\)
  2. \(~~x~~\) \(~~1~~\) \(~~2~~\) \(~~3~~\) \(~~4~~\)
    \(\left(\dfrac{5}{4}\right)^x\) \(\) \(\) \(\) \(\)
Solution

  1. \(~~x~~\) \(~~1~~\) \(~~2~~\) \(~~3~~\) \(~~4~~\)
    \(\left(\dfrac{2}{3}\right)^x\) \(\dfrac{2}{3}\) \(\dfrac{4}{9}\) \(\dfrac{8}{27}\) \(\dfrac{16}{81}\)

    When we multiply a number by \(\dfrac{2}{3}\text{,}\) the product is smaller than the original number, so the powers of \(\dfrac{2}{3}\) decrease as the exponent increases. We can compare the powers more easily by converting the fractions to decimals, rounded to three places:

    \begin{equation*} \dfrac{2}{3} = 0.667,~\dfrac{4}{9} = 0.444,~\dfrac{8}{27} = 0.296,~\dfrac{16}{81} = 0.198 \end{equation*}

  2. \(~~x~~\) \(~~1~~\) \(~~2~~\) \(~~3~~\) \(~~4~~\)
    \(\left(\dfrac{5}{4}\right)^x\) \(\dfrac{5}{4}\) \(\dfrac{25}{16}\) \(\dfrac{125}{64}\) \(\dfrac{625}{256}\)

    When we multiply a number by \(\dfrac{5}{4}\text{,}\) the product is larger than the original number, so the powers of \(\dfrac{5}{4}\) increase as the exponent increases. We can compare the powers more easily by converting the fractions to decimals, rounded to three places:

    \begin{equation*} \dfrac{5}{4} = 1.25,~\dfrac{25}{16} = 1.563,~\dfrac{125}{64} = 1.953,~\dfrac{625}{256} = 2.441 \end{equation*}
Example 4.10.

Complete the table of powers. As the exponent increases, do the powers increase or decrease?

  1. \(~~x~~\) \(~~1~~\) \(~~2~~\) \(~~3~~\) \(~~4~~\)
    \(0.2^x\) \(\) \(\) \(\) \(\)
  2. \(~~x~~\) \(~~1~~\) \(~~2~~\) \(~~3~~\) \(~~4~~\)
    \(1.2^x\) \(\) \(\) \(\) \(\)
Solution
  1. \(~~x~~\) \(~~1~~\) \(~~2~~\) \(~~3~~\) \(~~4~~\)
    \(0.2^x\) \(0.2\) \(0.04\) \(0.008\) \(0.0016\)
    The powers decrease.
  2. \(~~x~~\) \(~~1~~\) \(~~2~~\) \(~~3~~\) \(~~4~~\)
    \(1.2^x\) \(1.2\) \(1.44\) \(1.728\) \(2.0736\)
    The powers increase.

Subsubsection Exercise

Complete the table of powers. As the exponent increases, do the powers increase or decrease?

  1. \(~~x~~\) \(~~1~~\) \(~~2~~\) \(~~3~~\) \(~~4~~\)
    \(\left(\dfrac{3}{4}\right)^x\) \(\) \(\) \(\) \(\)
  2. \(~~x~~\) \(~~1~~\) \(~~2~~\) \(~~3~~\) \(~~4~~\)
    \(\left(\dfrac{4}{3}\right)^x\) \(\) \(\) \(\) \(\)
Answer
  1. \(~~x~~\) \(~~1~~\) \(~~2~~\) \(~~3~~\) \(~~4~~\)
    \(\left(\dfrac{3}{4}\right)^x\) \(\dfrac{3}{4}\) \(\dfrac{9}{16}\) \(\dfrac{27}{64}\) \(\dfrac{81}{256}\)
    Decrease
  2. \(~~x~~\) \(~~1~~\) \(~~2~~\) \(~~3~~\) \(~~4~~\)
    \(\left(\dfrac{4}{3}\right)^x\) \(\dfrac{4}{3}\) \(\dfrac{16}{9}\) \(\dfrac{64}{27}\) \(\dfrac{257}{81}\)
    Increase

Complete the table of powers. As the exponent increases, do the powers increase or decrease?

  1. \(~~x~~\) \(~~1~~\) \(~~2~~\) \(~~3~~\) \(~~4~~\)
    \(0.8^x\) \(\) \(\) \(\) \(\)
  2. \(~~x~~\) \(~~1~~\) \(~~2~~\) \(~~3~~\) \(~~4~~\)
    \(1.5^x\) \(\) \(\) \(\) \(\)
Answer
  1. \(~~x~~\) \(~~1~~\) \(~~2~~\) \(~~3~~\) \(~~4~~\)
    \(0.8^x\) \(0.8\) \(0.64\) \(0.512\) \(0.4096\)
    Decrease
  2. \(~~x~~\) \(~~1~~\) \(~~2~~\) \(~~3~~\) \(~~4~~\)
    \(1.5^x\) \(1.5\) \(2.25\) \(3.375\) \(5.0625\)
    Increase