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Elementary Algebra

Katherine Yoshiwara

Section 9.1 Laws of Exponents

Subsection Products and Quotients

In Chapter 7 we learned the first and second laws of exponents, which we use to compute products and quotients of powers.

Example 9.1.

Simplify each product or quotient.
  1. \(\displaystyle 5^2 \cdot 5^6\)
  2. \(\displaystyle \dfrac{3^7}{3^2}\)
  3. \(\displaystyle \dfrac{2^3}{2^5}\)

Solution.

  1. To multiply two powers with the same base, we add the exponents and leave the base unchanged.
    \begin{equation*} 5^2 \cdot 5^6 = 5^8 \end{equation*}
  2. To divide two powers with the same base, we subtract the smaller exponent from the larger. If the larger exponent occurs in the numerator, we put the power in the numerator.
    \begin{equation*} \dfrac{3^7}{3^2} = 3^5~~~~~~~~\blert{\text{Larger exponentis in the numerator.}} \end{equation*}
  3. If the larger exponent occurs in the denominator, we put the power in the denominator.
    \begin{equation*} \dfrac{2^3}{2^5} = \dfrac{1}{2^2}~~~~~~~~\blert{\text{Larger exponent is in the denominator.}} \end{equation*}

Caution 9.2.

In Example 9.1a, it is not correct to multiply the bases:
\begin{equation*} 5^2 \cdot 5^6 \rightarrow 25^8~~~~~~~~\alert{\text{Incorrect!}} \end{equation*}
The exponents tell us how many copies of the base to multiply together, but the base does not change when we apply the first law of exponents.

QuickCheck 9.3.

Why do we add exponents when we multiply powers with the same base?
Answer.
The exponents tell us how many copies of the base to multiply together.
These laws are expressed in symbols below.

Laws of Exponents.

First Law of Exponents
\begin{equation*} \blert{a^m \cdot a^n = a^{m+n}} \end{equation*}
Second Law of Exponents
\begin{gather*} \blert{\dfrac{a^m}{a^n} = a^{m-n}~~~~(n \lt m)}\\ \blert{\dfrac{a^m}{a^n} = \dfrac{1}{a^{n-m}}~~~~(n \gt m)} \end{gather*}
In this Lesson we’ll study three more laws of exponents.

Subsection Power of a Power

Consider the expression \(~(x^4)^3\text{,}\) or the cube of \(x^4\text{.}\) We can simplify this expression as follows.
\begin{equation*} (x^4)^3 = (x^4)(x^4)(x^4) = x^{4+4+4} = x^{12}~~~~~~\blert{\text{Add exponents.}} \end{equation*}
We wrote \(~(x^4)^3~\) as a repeated product and applied the first law of exponents to add the exponents. Of course, because repeated addition is actually multiplication, we can just multiply the exponents together: \(~3(4)=12\text{.}\) This gives us another rule.

Third Law of Exponents.

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

Example 9.4.

Simplify each power of a power.
  1. \(\displaystyle (b^2)^4 = b^{2 \cdot 4} = b^8~~~~~~~~~~~~\blert{\text{Multiply exponents.}}\)
  2. \(\displaystyle (y^2)^5 = y^{2 \cdot 5} = y^{10}\)

Caution 9.5.

Note carefully the difference between the two expressions \(~(x^3)(x^4)~\) and \(~(x^3)^4\text{:}\)
\begin{align*} ~~~~(x^3)(x^4) = x^{\blert{3+4}} = x^7~~~~~~~~~ \amp \blert{\text{Add the exponents.}}\\ (x^3)^4 = x^{\blert{3\cdot 4}} = x^{12}~~~~~~~\amp\blert{\text{Multiply the exponents.}} \end{align*}
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.

QuickCheck 9.6.

Which is larger, \(~(2^3)^3~\) or \(~2^3 \cdot 2^3\text{?}\) Why?
Answer.
\(~(2^3)^3~\) is larger because it is a product of three copies of \((2^3).\)

Subsection Power of a Product

To simplify the expression \(~(2x)^3~\text{,}\) we can use the commutative and associative properties of multiplication to write
\begin{align*} (2x)^3 \amp = (2x)(2x)(2x)\\ \amp = 2 \cdot 2 \cdot 2 \cdot x \cdot x \cdot x\\ \amp = 2^3x^3 \end{align*}
Thus, to raise a product to a power we can raise each factor to the power. This example illustrates the following rule.

Fourth Law of Exponents.

To raise a product to a power, raise each factor to the power. In symbols,
\begin{equation*} \blert{(ab)^n = a^nb^n} \end{equation*}

Example 9.7.

Simplify each power of a product.
  1. \(\displaystyle (5ab)^2 = 5^2a^2b^2 = 25a^2b^2~~~~~~~~\blert{\text{Square each factor.}}\)
  2. \begin{align*} (-xy^3)^4 \amp = (-x)^4(y^3)^4 \amp \amp \blert{\text{Raise each factor to the fourth power;}}\\ \amp = x^4y^{12} \amp \amp \blert{\text{apply the third law of exponents.}} \end{align*}

Caution 9.8.

Note the difference between the two expressions \(~3a^2~\) and \(~(3a)^2~\text{:}\)
\begin{gather*} 3a^2~~~~\text{cannot be simplified, but}\\ (3a)^2 = 3^2a^2 = 9a^2 \end{gather*}
In the expression \(~3a^2\text{,}\) only the factor \(a\) is squared, but in \(~(3a)^2~\) both \(3\) and \(a\) are squared.

QuickCheck 9.9.

Explain why we cannot use the fourth law of exponents to simpify \(~(a+b)^3~\) as \(~a^3+b^3~\text{.}\)
Answer.
Because \(~a+b~\) is a sum, not a product.

Subsection Power of a Quotient

To simplify the expression \(\left(\dfrac{x}{2}\right)^4\text{,}\) we multiply together four copies of the fraction \(\dfrac{x}{2}\text{.}\) That is,
\begin{align*} \left(\dfrac{x}{2}\right)^4 \amp = \dfrac{x}{2} \cdot \dfrac{x}{2} \cdot \dfrac{x}{2} \cdot \dfrac{x}{2} = \dfrac{x \cdot x \cdot x \cdot x}{2 \cdot 2 \cdot 2 \cdot 2}\\ \amp = \dfrac{x^4}{2^4} = \dfrac{x^4}{16} \end{align*}
This example suggests the following rule.

Fifth Law of Exponents.

To raise a quotient to a power, raise both the numerator and the denominator to the power. In symbols,
\begin{equation*} \blert{\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}} \end{equation*}

Example 9.10.

Simplify each power of a quotient.
  1. \(\displaystyle \left(\dfrac{x}{y}\right)^5 = \dfrac{x^5}{y^5} ~~~~~~~~~~ \blert{\text{Raise top and bottom to the fifth power.}}\)
  2. \begin{align*} \left(\dfrac{2}{y^2}\right)^3 \amp = \dfrac{2^3}{(y^2)^3} \amp \amp \blert{\text{Raise top and bottom to the third power.}}\\ \amp =\dfrac{2^3}{y^{2 \cdot 3}} = \dfrac{8}{y^6} \amp \amp \blert{\text{Apply the third law of exponents.}} \end{align*}

QuickCheck 9.11.

Why does \(\left(\dfrac{1}{x}\right)^4 = \dfrac{1}{x^4}\text{?}\)
Answer.
Because \(1^4=1\text{.}\)

Subsection Using the Laws of Exponents

We can use the laws of exponents along with the order of operations to simplify algebraic expressions. The five laws of exponents are stated together below. All of the laws are valid when the base is not equal to zero and when the exponents \(m\) and \(n\) are positive integers.

Laws of Exponents.

  1. \(\displaystyle \blert{~a^m \cdot a^n = a^{m+n}}\)
  2. \(\blert{~\dfrac{a^m}{a^n} = a^{m-n}~~~~(n \lt m)}\)
    \(\blert{~\dfrac{a^m}{a^n} = \dfrac{1}{a^{n-m}}~~~~(n \gt m)}\)
  3. \(\displaystyle \blert{~(a^m)^n = a^{mn}}\)
  4. \(\displaystyle \blert{~(ab)^n = a^nb^n}\)
  5. \(\displaystyle \blert{~\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}}\)

Example 9.12.

Simplify \(~~2x^2y(3xy^2)^4\)

Solution.

According to the order of operations, we should perform powers before products, so we first simplify the factor \((3xy^2)^4\text{.}\)
\begin{align*} (3xy^2)^4 \amp = 3^4x^4(y^2)^4 \amp \amp \blert{\text{Fourth law: raise each factor to the power.}}\\ \amp = 81x^4y^8 \amp \amp \blert{\text{Third law:}~(y^2)^4=y^{2 \cdot 4}} \end{align*}
Now multiply the result by \(~2x^2y~\) and apply the first law to obtain
\begin{align*} 2x^2y (81x^4y^8)\amp = 2 \cdot 81x^2 \cdot x^4 \cdot y \cdot y^8\\ \amp = 162x^6y^9 \end{align*}

QuickCheck 9.13.

In Example 9.12, why did we start by simplifying \(~(3xy^2)^4\text{?}\)
Answer.
Because we should perform powers before products.

Example 9.14.

Simplify \(~~(-x)^4(-xy)^3\)

Solution.

Each power should be simplified before we compute their product. Because \(4\) is an even exponent, \(~(-x)^4~\) is positive, so
\begin{equation*} (-x)^4 = x^4 \end{equation*}
To simplify \((-xy)^3\text{,}\) we apply the fourth law of exponents:
\begin{align*} (-xy)^3 \amp = (-x)^3y^3 ~~~~\blert{\text{3 is an odd power, so} ~(-x)^3 = -x^3.}\\ \amp = -x^3y^3 \end{align*}
Finally, we multiply the powers together to get
\begin{align*} (-x)^4(-xy)^3 \amp = x^4(-x^3y^3)~~~~\blert{\text{Apply the first law.}}\\ \amp = -x^7y^3 \end{align*}

QuickCheck 9.15.

In the expression \(~-x^3y^3~\text{,}\) does the negtive sign apply to both factors, or only one of them?
Answer.
Only one of them.

Skills Warm-Up 9.16.

Simplify each expression according to the order of operations.
  1. \(\displaystyle 10-6^2\)
  2. \(\displaystyle 10(-6^2)\)
  3. \(\displaystyle (10-6)^2\)
  4. \(\displaystyle 2 \cdot 5^2-3^2\)
  5. \(\displaystyle -2 \cdot 5^2 (-3^2)\)
  6. \(\displaystyle -2^2-(5-3)^2\)

Answer.

  1. \(\displaystyle -26\)
  2. \(\displaystyle 360\)
  3. \(\displaystyle 16\)
  4. \(\displaystyle 41\)
  5. \(\displaystyle 450\)
  6. \(\displaystyle -8\)

Subsection Lesson

Activity 9.1. Laws of Exponents.

  1. Use the laws of exponents to simplify.
    1. \(\displaystyle (4x^4)(5x^5)\)
    2. \(\displaystyle \dfrac{8x^8}{4x^4}\)
    3. \(\displaystyle \dfrac{6a^3(a^4)}{9a^6}\)
  2. Use the laws of exponents to simplify.
    1. \(\displaystyle (y^2)^5\)
    2. \(\displaystyle (5^3)^6\)
    3. \(\displaystyle (n^4)^2\)
  3. Simplify using the laws of exponents. State the law you used in each case.
    1. \(\displaystyle (5^4)^2\)
    2. \(\displaystyle (5^4)(5^2)\)
  4. Use the laws of exponents to simplify.
    1. \(\displaystyle (6a^3q^6)^3\)
    2. \(\displaystyle \left(\dfrac{n^3}{k^4}\right)^8\)
    3. \(\displaystyle \dfrac{3^2a^3(3a^2)}{2^3a(2^3a^4)}\)

Activity 9.2. Simplifying Expressions.

  1. Recall that \(-x\) means \(-1 \cdot x\text{.}\) Use this idea to simplify each expression if possible.
    1. \(\displaystyle -x^4\)
    2. \(\displaystyle (-x)^4\)
    3. \(\displaystyle (-2)^4\)
    4. \(\displaystyle -x^5\)
    5. \(\displaystyle (-x)^5\)
    6. \(\displaystyle (-2)^5\)
  2. In general, an odd power of a negative number is , and an even power of a negative number is .
  3. Follow the steps to simplify \(~~\left(\dfrac{-2ab^4}{3c^5}\right)^3\)
    \(\blert{\text{Step 1 Apply the fifth law: raise numerator and denominator}}\)
    \(\blert{\text{to the third power.}}\)
    \(\blert{\text{Step 2 Apply the fourth law: raise each factor to the third power.}}\)
    \(\blert{\text{Step 3 Apply the third law: simplify each power of a power.}}\)
  4. Follow the steps to simplify \(~~3x(xy^3)^2-xy^3+4x^3(y^2)^3\)
    \(\blert{\text{Step 1 Underline each term of the expression separately.}}\)
    \(\blert{\text{Step 2 Simplify the first term: apply the fourth law, then the third law.}}\)
    \(\blert{\text{Step 3 Simplify the last term: apply the third law.}}\)
    \(\blert{\text{Step 4 Add or subtract any like terms.}}\)

Subsubsection Wrap-Up

Objectives.
In this Lesson we practiced the following skills:
  • Applying the laws of exponents
  • Simplifying expressions involving powers
Questions.
  1. In Activity 1, explain the difference between \(\dfrac{8}{4}\) and \(\dfrac{x^8}{x^4}\text{.}\)
  2. Explain the difference between \(4^2\) and \((x^4)^2\text{.}\)
  3. Explain why \((5^4)(5^2)\) is not equal to \(25^6\text{.}\)

Activity 9.3. Homework Preview.

  1. Simplify.
    1. \(\displaystyle 3^6 \cdot 3^9\)
    2. \(\displaystyle \dfrac{3^6}{3^9}\)
    3. \(\displaystyle (3^6)^9\)
  2. Simplify.
    1. \(\displaystyle 2ab^2(2ab)^2\)
    2. \(\displaystyle \dfrac{4ab^2}{(4ab)^2}\)
    3. \(\displaystyle \dfrac{(3a^3b^2)^3}{3ab^3(-ab^3)}\)
  3. Simplify.
    1. \(\displaystyle 6-2np^3+np(p^2-n^2)\)
    2. \(\displaystyle 4np(-2np)-2np(-2np)^2\)
Answers to Homework Preview
    1. \(\displaystyle 3^{15}\)
    2. \(\displaystyle \dfrac{1}{3^3}\)
    3. \(\displaystyle 3^{54}\)
    1. \(\displaystyle 8a^3b^4\)
    2. \(\displaystyle \dfrac{1}{4a^2}\)
    3. \(\displaystyle -9a^7\)
    1. \(\displaystyle 6-np^3-n^3p\)
    2. \(\displaystyle -8n^2p^2-8n^3p^3\)

Exercises Homework 9.1

1.

Simplify each power by using the third law of exponents.
  1. \(\displaystyle \left(t^3\right)^5\)
  2. \(\displaystyle \left(b^4\right)^2\)
  3. \(\displaystyle \left(w^{12}\right)^{12}\)

2.

Simplify each power by using the fourth law of exponents.
  1. \(\displaystyle (5x)^2\)
  2. \(\displaystyle (-3wz)^4\)
  3. \(\displaystyle (-ab)^5\)

3.

Simplify each power by using the fifth law of exponents.
  1. \(\displaystyle \left(\dfrac{w}{2}\right)^6\)
  2. \(\displaystyle \left(\dfrac{5}{v}\right)^4\)
  3. \(\displaystyle \left(\dfrac{-m}{p}\right)^3\)

4.

Simplify each expression.
  1. \(\displaystyle x^3 \cdot x^6\)
  2. \(\displaystyle (x^3)^6\)
  3. \(\displaystyle \dfrac{x^3}{x^6}\)
  4. \(\displaystyle \dfrac{x^6}{x^3}\)

5.

Explain the difference between the first and third laws of exponents. Use examples.

6.

Find and correct the error in each calculation.
  1. \(\displaystyle 2 \cdot 3^2 \rightarrow 36\)
  2. \(\displaystyle -10^2 \rightarrow 100\)
  3. \(\displaystyle a^4 \cdot a^3 \rightarrow a^{12}\)

Exercise Group.

For Problems 7–9, use the laws of exponents to simplify the expression.
7.
\((2p^3)^5\)
8.
\(\left(\dfrac{-3}{q^4}\right)^5\)
9.
\(\left(\dfrac{-2h^2}{m^3}\right)^4\)

Exercise Group.

For Problems 10–15, simplify.
10.
\(x^3(x^2)^5\)
11.
\((2x^3y)^2(xy^3)^4\)
12.
\(\left[ab^2\left(a^2b\right)^3\right]^3\)
13.
\(-a^2(-a)^2\)
14.
\(-(-xy)^2(xy^2)\)
15.
\(-4p\left(-p^2q^2\right)^2\left(-q^3\right)^2\)

Exercise Group.

For Problems 16–18, simplify.
16.
\(2y(y^3)^2-2y^4(3y)^3\)
17.
\(2a(a^2)^4+3a^2(a^6)-a^2(a^2)^3\)
18.
\(-3v^2(2v^3-v^2)+v(-4v)^2\)

Exercise Group.

For Problems 19–23, simplify each pair of expressions as much as possible.
19.
  1. \(\displaystyle 4x^2+2x^4\)
  2. \(\displaystyle 4x^2(2x^4)\)
20.
  1. \(\displaystyle (-x)^3x^4\)
  2. \(\displaystyle [(-x^3)(-x)]^4\)
21.
  1. \(\displaystyle (3x^2)^4(2x^4)^2\)
  2. \(\displaystyle (3x^2)^4-(2x^4)^2\)
22.
  1. \(\displaystyle 6x^3-3x^6\)
  2. \(\displaystyle 6x^3(-3x^6)\)
23.
  1. \(\displaystyle 6x^3-3x^3(x^3)\)
  2. \(\displaystyle (6x^3-3x^3)x^3\)

Exercise Group.

For Problems 24–26, find the value of \(n\text{.}\)
24.
\(b^3 \cdot b^n = b^9\)
25.
\(\dfrac{c^8}{c^n} = c^2\)
26.
\(\dfrac{n^3}{3^3} = 8\)

Exercise Group.

For Problems 27–28, factor.
27.
\(x^4+x^6 = x^2(\fillinmath{XXXXXX})\)
28.
\(4m^4-4m^8+8m^{16}=4m^4(\fillinmath{XXXXXX})\)

Exercise Group.

Mental Exercise: For Problems 29–34, replace the comma with the appropriate symbol, \(\lt\text{,}\) \(\gt\text{,}\) or \(=\text{.}\) Do not use pencil, paper, or calculator.
29.
\(-8^2, ~64\)
30.
\((-3)^5, ~-3^5\)
31.
\(\left(\dfrac{-7}{4}\right)^{11}, ~0\)
32.
\(6^{10} \cdot 4^{10}, ~24^{10}\)
33.
\((8-2)^3, ~8-2^3\)
34.
\((17^4)^5, ~(17^5)^4\)
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