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Section 6.1 Integer Exponents

Subsection 1. Use the laws of exponents

Recall the five Laws of Exponents.

Laws of Exponents.

  1. \(\displaystyle \displaystyle{a^m\cdot a^n = a^{m+n}}\)

  2. \(\displaystyle \dfrac{a^m}{a^n}=\begin{cases} a^{m-n} \amp \text{if}~m \gt n\\ \dfrac{1}{a^{n-m}} \amp \text{if}~n \gt m \end{cases}\)

  3. \(\displaystyle \displaystyle{\left(a^m\right)^n=a^{mn}}\)

  4. \(\displaystyle \displaystyle{\left(ab\right)^n=a^n b^n}\)

  5. \(\displaystyle \displaystyle{\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}}\)

Subsubsection Examples

Example 6.1.

Multiply \(~(2x^2y)(5x^4y^3)~\text{.}\)

Solution

Rearrange the factors to group together the numerical coefficients and the powers of each base.

\begin{equation*} (2x^2y)(5x^4y^3) = (2)(5)x^2x^4yy^3 \end{equation*}

Multiply the coefficients together, and use the first law of exponents to find the products of the variable factors.

\begin{equation*} (2)(5)x^2x^4yy^3 = 10x^6y^4 ~~~~~~~~ \blert{\text{Add exponents on each base.}} \end{equation*}
Example 6.2.

Divide \(~\dfrac{3x^2y^4}{6x^3y}\text{.}\)

Solution

Consider the numerical coefficients and the powers of each base separately. Use the second law of exponents to simplify each quotient of powers.

\begin{align*} \dfrac{3x^2y^4}{6x^3y} \amp = \dfrac{3}{6} \cdot \dfrac{x^2}{x^3} \cdot \dfrac{y^4}{y} \amp\amp \blert{\text{Subtract exponents on each base.}}\\ \amp = \dfrac{1}{2} \cdot \dfrac{1}{x^{3-2}} \cdot y^{4-1} \amp\amp \blert{\text{Multiply factors.}}\\ \amp = \dfrac{1}{2} \cdot \dfrac{1}{x} \cdot \dfrac{y^3}{1} = \dfrac{y^3}{2x} \end{align*}
Example 6.3.

Simplify \(~(5a^3b)^2\text{.}\)

Solution

Apply the fourth law of exponents and square each factor.

\begin{equation*} (5a^3b)^2 = 5^2(a^3)^2b^2 = 25a^6b^2 ~~~~~~~~ \blert{\text{Apply the third law: multiply exponents.}} \end{equation*}
Example 6.4.

Simplify \(~\left(\dfrac{2}{y^3}\right)^3\text{.}\)

Solution

Apply the fifth law of exponents.

\begin{align*} \left(\dfrac{2}{y^3}\right)^3 \amp = \dfrac{2^3}{(y^3)^3} \amp\amp \blert{\text{Cube numerator and denominator.}}\\ \amp = \dfrac{2^3}{y^{3(3)}} = \dfrac{8}{y^9} \amp\amp \blert{\text{Apply the third law.}} \end{align*}

Subsubsection Exercises

Multiply \(~-3a^4b(-4a^3b)\text{.}\)

Answer
\(12a^7b^2\)

Divide \(~\dfrac{8x^2y}{12x^5y^3}\text{.}\)

Answer
\(\dfrac{2}{3x^3y^2}\)

Simplify \(~(6pq^4)^3\text{.}\)

Answer
\(216p^3q^{12}\)

Simplify \(~\left(\dfrac{n^3}{k^4}\right)^8\text{.}\)

Answer
\(\dfrac{n^{24}}{k^{32}}\)

Subsection 2. Evaluate powers with negative exponents

Remember that a negative exponent indicates a reciprocal, so for example

\begin{equation*} x^{-2} = \dfrac{1}{x^2} \end{equation*}

A negative exponent does not mean that the power is negative. So for example

\begin{equation*} 4^{-2} = \dfrac{1}{16}\text{;} \end{equation*}

\(4^{-2}\) does not mean \(-16\text{.}\)

Subsubsection Examples

Example 6.9.

Write each expression without using negative exponents.

  1. \(\displaystyle 10^{-4}\)
  2. \(\displaystyle \left(\dfrac{x}{4}\right)^{-3}\)
Solution
  1. \(10^{-4} = \dfrac{1}{10^4} = \dfrac{1}{10,000}~\text{,}\) or \(~0.0001\text{.}\)

  2. To compute a negative power of a fraction, we compute the corresponding positive power of its reciprocal. Thus,

    \begin{equation*} \left(\dfrac{x}{4}\right)^{-3} = \left(\dfrac{4}{x}\right)^3 = \dfrac{64}{x^3} \end{equation*}
Example 6.10.

Write each expression using negative exponents.

  1. \(\displaystyle \dfrac{1}{3a^4a^2}\)
  2. \(\displaystyle \dfrac{8}{x^4}\)
Solution
  1. \(\displaystyle \dfrac{1}{3a^4a^2} = 3^{-4}a^{-2}\)
  2. \(\displaystyle \dfrac{8}{x^4} = 8x^{-4}\)

Subsubsection Exercises

Write each expression using negative exponents and evaluate.

  1. \(\displaystyle (-6)^{-2}\)
  2. \(\displaystyle \left(\dfrac{3}{5}\right)^{-2}\)
Answer
  1. \(\displaystyle \dfrac{1}{6^2} = \dfrac{1}{36}\)
  2. \(\displaystyle \dfrac{5^2}{3^2} = \dfrac{25}{9}\)

Write each expression using negative exponents.

  1. \(\displaystyle 4t^{-2}\)
  2. \(\displaystyle (4t)^{-2}\)
Answer
  1. \(\displaystyle \dfrac{4}{t^2}\)
  2. \(\displaystyle \dfrac{1}{16t^2}\)

Subsection 3. Use scientific notation

If we move the decimal point to the left, we are making a number smaller, so we must multiply by a positive power of 10 to compensate. If we move the decimal point to the right, we must multiply by a negative power of 10.

Subsubsection Example

Example 6.13.

Write each number in scientific notation.

  1. \(\displaystyle 62,000,000\)
  2. \(\displaystyle 0.000431\)
Solution

  1. First, we position the decimal point so that there is just one nonzero digit to the left of the decimal.

    \begin{equation*} 62,000,000 = 6.2 \times \underline{\hspace{2.727272727272727em}} \end{equation*}

    To recover \(62,000,000\) from \(6.2\text{,}\) we must move the decimal point seven places to the right. Therefore, we multiply \(6.2\) by \(10^7\text{.}\)

    \begin{equation*} 62,000,000 = 6.2 \times 10^7 \end{equation*}

  2. First, we position the decimal point so that there is just one nonzero digit to the left of the decimal.

    \begin{equation*} 0.000431 = 4.31 \times \underline{\hspace{2.727272727272727em}} \end{equation*}

    To recover \(0.000431\) from \(4.31\text{,}\) we must move the decimal point seven places to the right. Therefore, we multiply \(4.31\) by \(10^{-4}\text{.}\)

    \begin{equation*} 0.000431 = 4.31 \times 10^{-4} \end{equation*}

Subsubsection Exercise

Write each number in scientific notation.

  1. The largest living animal is the blue whale, with an average weight of \(~120,000,000~\) grams.

  2. The smallest animal is the fairy fly beetle, which weighs about \(~0.000005~\) grams.

Answer
  1. \(\displaystyle 1.2 \times 10^8\)
  2. \(\displaystyle 5 \times 10^{-6}\)