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Modeling, Functions, and Graphs

Section A.6 Laws of Exponents

In this section, we review the rules for performing operations on powers.

Subsection Product of Powers

Consider a product of two powers with the same base.
\begin{equation*} (a^3) (a^2) = a a a \cdot a a = a^5 \end{equation*}
because \(a\) occurs as a factor five times. The number of \(a\)’s in the product is the \(sum\) of the number of \(a\)’s in each factor.

First Law of Exponents: Product of Powers.

To multiply two powers with the same base, add the exponents and leave the base unchanged.
\begin{equation*} a^m \cdot a^n = a^{m+n} \end{equation*}

Example A.36.

figure showing powers
Here are some mistakes to avoid.

Caution A.37.

  1. Note that we do not multiply the exponents when simplifying a product. For example,
    \begin{equation*} b^4 \cdot b^2 \ne b^8 \end{equation*}
    You can check this with your calculator by choosing a value for \(b\text{,}\) for instance, \(b = 3\text{:}\)
    \begin{equation*} 3^4 \cdot 3^2\ne 3^8 \end{equation*}
  2. In order to apply the first law of exponents, the bases must be the same. For example,
    \begin{equation*} 2^3 \cdot 3^5 \ne 6^8 \end{equation*}
    (Check this on your calculator.)
  3. We do not multiply the bases when simplifying a product. In Example A.36a, note that
    \begin{equation*} 5^3 \cdot 5^4\ne 25^7 \end{equation*}
  4. Although we can simplify the product \(x^2x^3\) as \(x^5\text{,}\) we cannot simplify the sum \(x^2 + x^3\text{,}\) because \(x^2\) and \(x^3\) are not like terms.

Example A.38.

Multiply \((-3x^4z^2) (5x^3 z)\text{.}\)
Rearrange the factors to group the numerical coefficients and the powers of each base. Apply the first law of exponents.
\begin{align*} (-3x^4z^2) (5x^3z) \amp = (-3) (5)x^4x^3z^2z\\ \amp = -15x^7z^3 \end{align*}

Subsection Quotients of Powers

To reduce a fraction, we divide both numerator and denominator by any common factors.
\begin{equation*} \frac{x^7}{x^4}=\frac{xxx\Ccancel[blue]{x}\Ccancel[blue]{x}\Ccancel[blue]{x}\Ccancel[blue]{x}}{\Ccancel[blue]{x}\Ccancel[blue]{x}\Ccancel[blue]{x}\Ccancel[blue]{x}}=\frac{x^3}{1}=x^3 \end{equation*}
We can obtain the same result more quickly by subtracting the exponent of the denominator from the exponent of the numerator.
\begin{equation*} \frac{x^7}{x^4}=x^{7-4}=x^3 \end{equation*}
What if the larger power occurs in the denominator of the fraction?
\begin{equation*} \frac{x^4}{x^7}=\frac{\Ccancel[blue]{x}\Ccancel[blue]{x}\Ccancel[blue]{x}\Ccancel[blue]{x}}{xxx\Ccancel[blue]{x}\Ccancel[blue]{x}\Ccancel[blue]{x}\Ccancel[blue]{x}}=\frac{1}{x^3} \end{equation*}
In this case, we subtract the exponent of the numerator from the exponent of the denominator.
\begin{equation*} \frac{x^4}{x^7}=\frac{1}{x^{7-4}}=\frac{1}{x^3} \end{equation*}
These examples suggest the following law.

Second Law of Exponents: Quotient of Powers.

To divide two powers with the same base, subtract the smaller exponent from the larger one, keeping the same base.
  1. If the larger exponent occurs in the numerator, put the power in the numerator.
    \begin{equation*} \text{If }m\gt n,~\text{ then }~ \frac{a^m}{a^n}=a^{m-n} \hphantom{blank}(a \ne 0) \end{equation*}
  2. If the larger exponent occurs in the denominator, put the power in the denominator.
    \begin{equation*} \text{If }m\lt n,~\text{ then }~ \frac{a^m}{a^n}=\frac{1}{a^{n-m}} \hphantom{blank}(a \ne 0) \end{equation*}

Example A.39.

  1. \(\displaystyle \dfrac{3^8}{3^2}=3^{8-2}=3^6\hphantom{blankblankbla}\blert{\text{Subtract exponents: }~8 \gt 2.}\)
  2. \(\displaystyle \displaystyle{\frac{w^3}{w^6}=\frac{1}{w^{6-3}}=\frac{1}{w^3}\hphantom{blankblank}\blert{\text{Subtract exponents: }3\lt 6.}}\)

Example A.40.

Divide \(~~\dfrac{3x^2 y^4}{6x^3 y}\)
Consider the numerical coefficients and the powers of each variable separately. Use the second law of exponents to simplify each quotient of powers.
\begin{align*} \frac{3x^2 y^4}{6x^3 y} \amp = \frac{3}{6}\cdot \frac{x^2}{x^3} \cdot \frac{y^4}{y}\amp\amp\blert{\text{Subtract exponents.}}\\ \amp = \frac{1}{2}\cdot \frac{1}{x^{3-2}} \cdot y^{4-1}\\ \amp = \frac{1}{2}\cdot \frac{1}{x} \cdot y^3=\frac{y^3}{2x} \end{align*}

Subsection Power of a Power

Consider the expression \(\left(a^4\right)^3\text{,}\) the third power of \(a^4\text{.}\)
\begin{equation*} \left(a^4\right)^3=\left(a^4\right)\left(a^4\right)\left(a^4\right) =a^{4+4+4}=a^{12} \hphantom{blankblank}\blert{\text{Add exponents.}} \end{equation*}
We can obtain the same result by multiplying the exponents together.
\begin{equation*} \left(a^4\right)^3=a^{4 \,\cdot\, 3}=a^{12} \end{equation*}

Third Law of Exponents: Power of a Power.

To raise a power to a power, keep the same base and multiply the exponents.
\begin{equation*} \left(a^m\right)^n = a^{mn} \end{equation*}

Example A.41.

multiplying exponents

Caution A.42.

Notice the difference between the expressions
\begin{equation*} (x^3) (x^4) = x^{3+4} = x^7 \end{equation*}
\begin{equation*} \left(x^3\right)^4 = x^{3\, \cdot \, 4} = x^{12} \end{equation*}
The first expression is a product, so we add the exponents. The second expression raises a power to a power, so we multiply the exponents.

Subsection Power of a Product

To simplify the expression \((5a)^3\text{,}\) we use the associative and commutative laws to regroup the factors as follows.
\begin{align*} (5a)^3 \amp = (5a) (5a) (5a)\\ \amp = 5 \cdot 5 \cdot 5 \cdot a \cdot a \cdot a\\ \amp = 5^3 a^3 \end{align*}
Thus, to raise a product to a power, we can simply raise each factor to the power.

Fourth Law of Exponents: Power of a Product.

A power of a product is equal to the product of the powers of each of its factors.
\begin{equation*} (ab)^n = a^n b^n \end{equation*}

Example A.43.

  1. \((5a)^3 = 5^3a^3 = 125a^3\hphantom{blank}\blert{\text{Cube each factor}}\text{.}\)
  2. \(\displaystyle \begin{aligned}[t] \left(-xy^2\right)^4\amp = (-x)^4\left(y^2\right)^4 \amp\amp\blert{\text{Raise each factor to the fourth power.}}\\ \amp =x^4y^8\amp\amp\blert{\text{Apply the third law of exponents.}} \end{aligned}\)

Caution A.44.

  1. Compare the two expressions \(3a^2\) and \((3a)^2\text{;}\) they are not the same. In the expression \(3a^2\text{,}\) only the factor \(a\) is squared. But in \((3a)^2\text{,}\) both \(3\) and \(a\) are squared. Thus,
    \begin{equation*} 3a^2 ~~\text{ cannot be simplified} \end{equation*}
    \begin{equation*} (3a)^2 = 3^2a^2 = 9a^2 \end{equation*}
  2. Compare the two expressions \((3a)^2\) and \((3 + a)^2\text{.}\) The fourth law of exponents applies to the product \(3a\text{,}\) but not to the sum \(3+a\text{.}\) Thus,
    \begin{equation*} (3 + a)^2 \ne 3^2 + a^2 \end{equation*}
    In order to simplify \((3 + a)^2\text{,}\) we must expand the binomial product:
    \begin{equation*} (3 + a)^2 = (3 + a) (3 + a) = 9 + 6a + a^2 \end{equation*}

Subsection Power of a Quotient

To simplify the expression \(\displaystyle{\left(\frac{x}{3}\right)^4} \text{,}\) we multiply together \(4\) copies of the fraction \(\dfrac{x}{3}\text{.}\)
\begin{align*} \left(\frac{x}{3}\right)^4 \amp = \frac{x}{3}\cdot \frac{x}{3}\cdot \frac{x}{3}\cdot \frac{x}{3} = \frac{x\cdot x\cdot x\cdot x}{3\cdot 3\cdot 3\cdot 3}\\ \amp = \frac{x^4}{3^4}=\frac{x^4}{81} \end{align*}
In general, we have the following rule.

Fifth Law of Exponents: Power of a Quotient.

To raise a quotient to a power, raise both the numerator and denominator to the power.
\begin{equation*} \left(\frac{a}{b}\right)^n=\frac{a^n}{b^n} \end{equation*}
For reference, we state all of the laws of exponents together. All the laws are valid when a and b are not equal to zero and when the exponents m and n are whole numbers.

Laws of Exponents.

  1. \(\displaystyle a^m\cdot a^n = a^{m+n}\)
    1. \(\displaystyle \dfrac{a^m}{a^n}=a^{m-n} \hphantom{blank}m\gt n\)
    2. \(\displaystyle \displaystyle{\frac{a^m}{a^n}=\frac{1}{a^{n-m}} \hphantom{blank}m\lt n}\)
  2. \(\displaystyle \left(a^m\right)^n=a^{m+n}\)
  3. \(\displaystyle (ab)^n=a^n b^n\)
  4. \(\displaystyle \displaystyle{\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} }\)

Example A.45.

Simplify \(~~5x^2 y^3\left(2xy^2\right)^4\)
According to the order of operations, we should perform any powers before multiplications. Thus, we begin by simplifying \((2xy^2)^4\text{.}\) We apply the fourth law.
\begin{align*} 5x^2 y^3\left(2xy^2\right)^4 \amp = 5x^2 y^3 \cdot 2^4x^4\left(y^2\right)^4 \amp\amp\blert{\text{Apply the fourth law.}}\\ \amp = 5x^2 y^3 \cdot 2^4x^4 y^8 \end{align*}
Finally, multiply powers with the same base. Apply the first law.
\begin{equation*} 5x^2 y^3 \cdot 2^4x^4 y^8 = 5 \cdot 2^4 x^2x^4 y^3 y^8 = 80x^6 y^{11} \end{equation*}

Example A.46.

Simplify \(~~\displaystyle{\left(\frac{2x}{z^2}\right)^3}\)
Begin by applying the fifth law.
\begin{align*} \left(\frac{2x}{z^2}\right)^3 \amp = \frac{(2x)^3}{\left(z^2\right)^2}\amp\amp \blert{\text{Apply the fourth law to the numerator and the third law to the denominator.}}\\ \amp = \frac{2^3x^3}{z^6}=\frac{8x^3}{z^6} \end{align*}

Subsection Section Summary

Subsubsection Vocabulary

Look up the definitions of new terms in the Glossary.
  • Exponent
  • Power

Subsubsection SKILLS

Practice each skill in the exercises listed.
  1. Apply the laws of exponents: #1–8
  2. Simplify expressions: #9–16, 25–32
  3. Multiply and divide power: #17–24

Exercises Exercises A.6

Exercise Group.

For Problems 1–8, simplify by applying the appropriate law of exponents.
  1. \(\displaystyle b^4\cdot b^5\)
  2. \(\displaystyle b^2\cdot b^8\)
  3. \(\displaystyle (q^3)(q)(q^5) \)
  4. \(\displaystyle (p^2)(p^4)(p^4) \)
  1. \(\displaystyle \dfrac{w^6}{w^3} \)
  2. \(\displaystyle \dfrac{c^{12}}{c^4} \)
  3. \(\displaystyle \dfrac{z^6}{z^9} \)
  4. \(\displaystyle \dfrac{b^4}{b^8} \)
  1. \(\displaystyle 2^7\cdot 2^2 \)
  2. \(\displaystyle 6^5\cdot 6^3 \)
  3. \(\displaystyle \dfrac{2^9}{2^4} \)
  4. \(\displaystyle \dfrac{8^6}{8^2} \)
  1. \(\displaystyle (d^3)^5 \)
  2. \(\displaystyle (d^4)^2 \)
  3. \(\displaystyle (5^4)^3 \)
  4. \(\displaystyle (4^3)^3 \)
  1. \(\displaystyle (6x)^3 \)
  2. \(\displaystyle (3y)^4 \)
  3. \(\displaystyle (2t^3)^5 \)
  4. \(\displaystyle (6s^2)^2 \)
  1. \(\displaystyle \left(\dfrac{w}{2} \right)^6 \)
  2. \(\displaystyle \left(\dfrac{5}{u} \right)^4 \)
  3. \(\displaystyle \left(\dfrac{-4}{p^5} \right)^3 \)
  4. \(\displaystyle \left(\dfrac{-3}{q^4} \right)^5 \)
  1. \(\displaystyle \left(\dfrac{h^2}{m^3} \right)^4 \)
  2. \(\displaystyle \left(\dfrac{n^3}{k^4} \right)^8 \)
  3. \(\displaystyle (-4a^2 b^4)^4 \)
  4. \(\displaystyle (-5ab^8)^3 \)
  1. \(\displaystyle \dfrac{ab^2}{(ab)^2} \)
  2. \(\displaystyle \dfrac{(x^2y)^2}{x^2y^2} \)
  3. \(\displaystyle \dfrac{(2mp)^3}{2m^3p} \)
  4. \(\displaystyle \dfrac{4^2 rt^4}{2^4r^4t} \)

Exercise Group.

For Problems 9–15, simplify if possible.
  1. \(\displaystyle w+w\)
  2. \(\displaystyle w(w)\)
  1. \(\displaystyle m^2-m^2\)
  2. \(\displaystyle m^2(-m^2)\)
  1. \(\displaystyle 4z^2 -6z^2 \)
  2. \(\displaystyle 4z^2(-6z^2) \)
  1. \(\displaystyle t^3+3t^3 \)
  2. \(\displaystyle t^3(3t^3) \)
  1. \(\displaystyle 4p^2+3p^3 \)
  2. \(\displaystyle 4p^2(3p^3) \)
  1. \(\displaystyle 2w^2 -5w^4 \)
  2. \(\displaystyle (2w^2)(-5w^4) \)
  1. \(\displaystyle 3^9\cdot3^8 \)
  2. \(\displaystyle 3^9+3^8 \)
  1. \(\displaystyle (-2)^7(-2)^5 \)
  2. \(\displaystyle -2^7 -2^5 \)

Exercise Group.

F mor Problems 17–20, multiply.
  1. \(\displaystyle (4y)(-6y) \)
  2. \(\displaystyle (-4z)(-8z) \)
  1. \(\displaystyle (2wz^3)(-8z) \)
  2. \(\displaystyle (4wz)(-9w^2z^2) \)
  1. \(\displaystyle -4x(3xy)(xy^3) \)
  2. \(\displaystyle (-5x^2)(2xy)(5x^2) \)
  1. \(\displaystyle -7ab^2(-3ab^3) \)
  2. \(\displaystyle -4a^2b(-3a^3b^2) \)

Exercise Group.

F dor Problems 21–22, divide.
  1. \(\displaystyle \dfrac{2a^3b}{8a^4b^5} \)
  2. \(\displaystyle \dfrac{8a^2b}{12a^5b^3} \)
  1. \(\displaystyle \dfrac{-12qw^4}{8qw^2} \)
  2. \(\displaystyle \dfrac{-12rz^6}{20rz} \)

Exercise Group.

For Problems 23–24, multiply or divide.
  1. \(\displaystyle \dfrac{-15bc(b^2c)}{-3b^3c^4} \)
  2. \(\displaystyle \dfrac{-25c(c^2d^2)}{-5c^8d^2} \)
  1. \(\displaystyle -2x^3(x^2y)(-4y^2) \)
  2. \(\displaystyle 3xy^3(-x^4)(-2y^2) \)

Exercise Group.

For Problems 25–28, simplify by applying the laws of exponents.
  1. \(\displaystyle b^3(b^2)^5 \)
  2. \(\displaystyle b(b^4)^6 \)
  1. \(\displaystyle (p^2q)^3(pq^3) \)
  2. \(\displaystyle (p^3)^4 (p^3q^4) \)
  1. \(\displaystyle (2x^3y)^2 (xy^3)^4 \)
  2. \(\displaystyle (3xy^2)^3 (2x^2y^2)^2 \)
  1. \(\displaystyle -a^2 (-a)^2 \)
  2. \(\displaystyle -a^3 (-a)^3 \)

Exercise Group.

For Problems 29–32, simplify by applying the laws of exponents.
  1. \(\displaystyle \left(\dfrac{-2x}{3y^2} \right)^3 \)
  2. \(\displaystyle \left(\dfrac{-x^2}{2y} \right)^4 \)
  1. \(\displaystyle \dfrac{(4x)^3}{(-2x^2)^2} \)
  2. \(\displaystyle \dfrac{(5x)^2}{(-3x^2)^3} \)
  1. \(\displaystyle \dfrac{(xy)^2(-x^2y)^3}{(x^2y^2)^2} \)
  2. \(\displaystyle \dfrac{(-x^2)(-x^2)^4}{(x^2)^3} \)
  1. \(\displaystyle \left(\dfrac{-2x}{y^2}\right)\left(\dfrac{y^2}{3x} \right)^2 \)
  2. \(\displaystyle \left(\dfrac{x^2 z}{2} \right)^3 \left(\dfrac{-2}{x^2z} \right)^3 \)