10.3: Use Multiplication Properties of Exponents (Part 2)
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Simplify Expressions Using the Product to a Power Property
We will now look at an expression containing a product that is raised to a power. Look for a pattern.
(2x)^{3}  
What does this mean?  2x • 2x • 2x 
We group the like factors together.  2 • 2 • 2 • x • x • x 
How many factors of 2 and of x?  2^{3} • x^{3} 
Notice that each factor was raised to the power.  (2x)^{3 }is 2^{3} • x^{3} 
We write:  $$\begin{split} &(2x)^{3} \\ &2^{3} \cdot x^{3} \end{split}$$ 
The exponent applies to each of the factors. This leads to the Product to a Power Property for Exponents.
Definition: Product to a Power Property of Exponents
If a and b are real numbers and m is a whole number, then
\[(ab)^{m} = a^{m} b^{m} \tag{10.2.27}\]
To raise a product to a power, raise each factor to that power.
An example with numbers helps to verify this property:
\[\begin{split} (2 \cdot 3)^{2} &\stackrel{?}{=} 2^{2} \cdot 3^{2} \\ 6^{2} &\stackrel{?}{=} 4 \cdot 9 \\ 36 &\stackrel{?}{=} 36\; \checkmark \end{split}\]
Example \(\PageIndex{10}\):
Simplify: (−11x)^{2}.
Solution
Use the Power of a Product Property, (ab)^{m} = a^{m} b^{m}.  $$(11)^{\textcolor{red}{2}} x^{\textcolor{red}{2}} \tag{10.2.28}$$ 
Simplify.  $$121x^{2} \tag{10.2.29}$$ 
Exercise \(\PageIndex{19}\):
Simplify: (−14x)^{2}.
 Answer

196x^{2}
Exercise \(\PageIndex{20}\):
Simplify: (−12a)^{2}.
 Answer

144a^{2}
Example \(\PageIndex{11}\):
Simplify: (3xy)^{3}.
Solution
Raise each factor to the third power.  $$3^{\textcolor{red}{3}} x^{\textcolor{red}{3}} y^{\textcolor{red}{3}} \tag{10.2.30}$$ 
Simplify.  $$27x^{3} y^{3} \tag{10.2.31}$$ 
Exercise \(\PageIndex{21}\):
Simplify: (−4xy)^{4}.
 Answer

256x^{4}y^{4}
Exercise \(\PageIndex{22}\):
Simplify: (6xy)^{3}.
 Answer

216x^{3}y^{3}
Simplify Expressions by Applying Several Properties
We now have three properties for multiplying expressions with exponents. Let’s summarize them and then we’ll do some examples that use more than one of the properties.
Definition: Properties of Exponents
If a, b are real numbers and m, n are whole numbers, then
Product Property  a^{m} • a^{n} = a^{m + n} 
Power Property  (a^{m})^{n} = a^{m • n} 
Product to a Power Property  (ab)^{m} = a^{m}b^{m} 
Example \(\PageIndex{12}\):
Simplify: (x^{2})^{6}(x^{5})^{4}.
Solution
Use the Power Property.  x^{12 }• x^{20} 
Add the exponents.  x^{32} 
Exercise \(\PageIndex{23}\):
Simplify: (x^{4})^{3}(x^{7})^{4}.
 Answer

x^{40}
Exercise \(\PageIndex{24}\):
Simplify: (y^{9})^{2}(y^{8})^{3}.
 Answer

y^{42}
Example \(\PageIndex{13}\):
Simplify: (−7x^{3}y^{4})^{2}.
Solution
Take each factor to the second power.  (−7)^{2}(x^{3})^{2}(y^{4})^{2} 
Use the Power Property.  49x^{6}y^{8} 
Exercise \(\PageIndex{25}\):
Simplify: (−8x^{4}y^{7})^{3}.
 Answer

512x^{12}y^{21}
Exercise \(\PageIndex{26}\):
Simplify: (−3a^{5}b^{6})^{4}.
 Answer

81a^{20}b^{24}
Example \(\PageIndex{14}\):
Simplify: (6n)^{2}(4n^{3}).
Solution
Raise 6n to the second power.  6^{2}n^{2} • 4n^{3} 
Simplify.  36n^{2} • 4n^{3} 
Use the Commutative Property.  36 • 4 • n^{2} • n^{3} 
Multiply the constants and add the exponents.  144n^{5} 
Notice that in the first monomial, the exponent was outside the parentheses and it applied to both factors inside. In the second monomial, the exponent was inside the parentheses and so it only applied to the n.
Exercise \(\PageIndex{27}\):
Simplify: (7n)^{2} (2n^{12}).
 Answer

98n^{14}
Exercise \(\PageIndex{28}\):
Simplify: (4m)^{2}(3m^{3}).
 Answer

48m^{5}
Example \(\PageIndex{15}\):
Simplify: (3p^{2}q)^{4}(2pq^{2})^{3}.
Solution
Use the Power of a Product Property.  3^{4}(p^{2})^{4}q^{4} • 2^{3}p^{3}(q^{2})^{3} 
Use the Power Property.  81p^{8}q^{4} • 8p^{3}q^{6} 
Use the Commutative Property.  81 • 8 • p^{8} • p^{3} • q^{4} • q^{6} 
Multiply the constants and add the exponents for each variable.  648p^{11}q^{10} 
Exercise \(\PageIndex{29}\):
Simplify: (u^{3}v^{2})^{5}(4uv^{4})^{3}.
 Answer

64u^{18}v^{22}
Exercise \(\PageIndex{30}\):
Simplify: (5x^{2}y^{3})^{2}(3xy^{4})^{3}.
 Answer

675x^{7}y^{18}
Multiply Monomials
Since a monomial is an algebraic expression, we can use the properties for simplifying expressions with exponents to multiply the monomials.
Example \(\PageIndex{16}\):
Multiply: (4x^{2})(−5x^{3}).
Solution
Use the Commutative Property to rearrange the factors.  4 • (−5) • x^{2} • x^{3} 
Multiply.  −20x^{5} 
Exercise \(\PageIndex{31}\):
Multiply: (7x^{7})(−8x^{4}).
 Answer

56x^{11}
Exercise \(\PageIndex{32}\):
Multiply: (−9y^{4})(−6y^{5}).
 Answer

54y^{9}
Example \(\PageIndex{17}\):
Multiply: \(\left(\dfrac{3}{4} c^{3} d\right)\)(12cd^{2}).
Solution
Use the Commutative Property to rearrange the factors.  \(\dfrac{3}{4}\) • 12 • c^{3} • c • d • d^{2} 
Multiply.  9c^{4}d^{3} 
Exercise \(\PageIndex{33}\):
Multiply: \(\left(\dfrac{4}{5} m^{4} n^{3} d\right)\)(15mn^{3}).
 Answer

12m^{5}n^{6}
Exercise \(\PageIndex{34}\):
Multiply: \(\left(\dfrac{2}{3} p^{5} q d\right)\)(18p^{6}q^{7}).
 Answer

12p^{11}q^{8}
Practice Makes Perfect
Simplify Expressions with Exponents
In the following exercises, simplify each expression with exponents.
 4^{5}
 10^{3}
 \(\left(\dfrac{1}{2}\right)^{2}\)
 \(\left(\dfrac{3}{5}\right)^{2}\)
 (0.2)^{3 }
 (0.4)^{3}
 (−5)^{4}
 (−3)^{5}
 −5^{4}
 −3^{5}
 −10^{4}
 −2^{6}
 \( \left(\dfrac{2}{3}\right)^{3}\)
 \( \left(\dfrac{1}{4}\right)^{4}\)
 −0.5^{2 }
 −0.1^{4}
Simplify Expressions Using the Product Property of Exponents
In the following exercises, simplify each expression using the Product Property of Exponents.
 x^{3} • x^{6}
 m^{4} • m^{2}
 a • a^{4}
 y^{12 }• y
 3^{5} • 3^{9}
 5^{10 }• 5^{6}
 z • z^{2} • z^{3}
 a • a^{3} • a^{5}
 x^{a} • x^{2}
 y^{p} • y^{3}
 y^{a} • y^{b}
 x^{p} • x^{q}
Simplify Expressions Using the Power Property of Exponents
In the following exercises, simplify each expression using the Power Property of Exponents.
 (u^{4})^{2}
 (x^{2})^{7}
 (y^{5})^{4}
 (a^{3})^{2}
 (10^{2})^{6}
 (2^{8})^{3}
 (x^{15})^{6}
 (y^{12})^{8}
 (x^{2})^{y}
 (y^{3})^{x}
 (5^{x})^{y}
 (7^{a})^{b}
Simplify Expressions Using the Product to a Power Property
In the following exercises, simplify each expression using the Product to a Power Property.
 (5a)^{2}
 (7x)^{2}
 (−6m)^{3}
 (−9n)^{3}
 (4rs)^{2}
 (5ab)^{3}
 (4xyz)^{4}
 (−5abc)^{3}
Simplify Expressions by Applying Several Properties
In the following exercises, simplify each expression.
 (x^{2})^{4} • (x^{3})^{2}
 (y^{4})^{3} • (y^{5})^{2}
 (a^{2})^{6} • (a^{3})^{8}
 (b^{7})^{5} • (b^{2})^{6}
 (3x)^{2}(5x)
 (2y)^{3} (6y)
 (5a)^{2}(2a)^{3}
 (4b)^{2}(3b)^{3}
 (2m^{6})^{3}
 (3y^{2})^{4}
 (10x^{2}y)^{3}
 (2mn^{4})^{5}
 (−2a^{3}b^{2})^{4}
 (−10u^{2}v^{4})^{3}
 \(\left(\dfrac{2}{3} x^{2} y \right)^{3}\)
 \(\left(\dfrac{7}{9} p q^{4} \right)^{2}\)
 (8a^{3})^{2}(2a)^{4}
 (5r^{2})^{3}(3r)^{2}
 (10p^{4})^{3}(5p^{6})^{2}
 (4x^{3})^{3}(2x^{5})^{4}
 \(\left(\dfrac{1}{2} x^{2} y^{3} \right)^{4}\) (4x^{5}y^{3})^{2}
 \(\left(\dfrac{1}{3} m^{3} n^{2} \right)^{4}\) (9m^{8}n^{3})^{2}
 (3m^{2}n)^{2}(2mn^{5})^{4}
 (2pq^{4})^{3}(5p^{6}q)^{2}
Multiply Monomials
In the following exercises, multiply the following monomials.
 (12x^{2})(−5x^{4})
 (−10y^{3})(7y^{2})
 (−8u^{6})(−9u)
 (−6c^{4})(−12c)
 \(\left(\dfrac{1}{5} r^{8} \right)\) (20r^{3})
 \(\left(\dfrac{1}{4} a^{5} \right)\) (36a^{2})
 (4a^{3}b)(9a^{2}b^{6})
 (6m^{4}n^{3})(7mn^{5})
 \(\left(\dfrac{4}{7} x y^{2} \right)\) (14xy^{3})
 \(\left(\dfrac{5}{8} u^{3} v \right)^{3}\) (24u^{5}v)
 \(\left(\dfrac{2}{3} x^{2} y \right) \left(\dfrac{3}{4} x y^{2} \right)\)
 \(\left(\dfrac{3}{5} m^{3} n^{2} \right) \left(\dfrac{5}{9} m^{2} n^{3} \right)\)
Everyday Math
 Email Janet emails a joke to six of her friends and tells them to forward it to six of their friends, who forward it to six of their friends, and so on. The number of people who receive the email on the second round is 6^{2}, on the third round is 6^{3}, as shown in the table. How many people will receive the email on the eighth round? Simplify the expression to show the number of people who receive the email.
Round  Number of people 

1  6 
2  6^{2} 
3  6^{3} 
…  … 
8  ? 
 Salary Raul’s boss gives him a 5% raise every year on his birthday. This means that each year, Raul’s salary is 1.05 times his last year’s salary. If his original salary was $40,000, his salary after 1 year was $40,000(1.05), after 2 years was $40,000(1.05)^{2}, after 3 years was $40,000(1.05)^{3}, as shown in the table below. What will Raul’s salary be after 10 years? Simplify the expression, to show Raul’s salary in dollars.
Year  Salary 

1  $40,000(1.05) 
2  $40,000(1.05)^{2} 
3  $40,000(1.05)^{3} 
…  … 
10  ? 
Writing Exercises
 Use the Product Property for Exponents to explain why x • x = x^{2}.
 Explain why −5^{3} = (−5)^{3} but −5^{4} ≠ (−5)^{4}.
 Jorge thinks \(\left(\dfrac{1}{2}\right)^{2}\) is 1. What is wrong with his reasoning?
 Explain why x^{3 }• x^{5} is x^{8}, and not x^{15}.
Self Check
(a) After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
(b) After reviewing this checklist, what will you do to become confident for all objectives?
Contributors and Attributions
Lynn Marecek (Santa Ana College) and MaryAnne AnthonySmith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at http://cnx.org/contents/fd53eae1fa2...49835c3c@5.191."