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Mathematics LibreTexts

10.2: Use Multiplication Properties of Exponents (Part 2)

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? 23 • x3
Notice that each factor was raised to the power. (2x)is 23 • x3
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 10.20:

Simplify: (−11x)2.

Solution
Use the Power of a Product Property, (ab)m = am bm. $$(-11)^{\textcolor{red}{2}} x^{\textcolor{red}{2}} \tag{10.2.28}$$
Simplify. $$121x^{2} \tag{10.2.29}$$

Exercise 10.39:

Simplify: (−14x)2.

Exercise 10.40:

Simplify: (−12a)2.

Example 10.21:

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 10.41:

Simplify: (−4xy)4.

Exercise 10.42:

Simplify: (6xy)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 am • an = am + n
Power Property (am)n = am • n
Product to a Power Property (ab)m = ambm

Example 10.22:

Simplify: (x2)6(x5)4.

Solution
Use the Power Property. x12 • x20
Add the exponents. x32

Exercise 10.43:

Simplify: (x4)3(x7)4.

Exercise 10.44:

Simplify: (y9)2(y8)3.

Example 10.23:

Simplify: (−7x3y4)2.

Solution
Take each factor to the second power. (−7)2(x3)2(y4)2
Use the Power Property. 49x6y8

Exercise 10.45:

Simplify: (−8x4y7)3.

Exercise 10.46:

Simplify: (−3a5b6)4.

Example 10.24:

Simplify: (6n)2(4n3).

Solution
Raise 6n to the second power. 62n2 • 4n3
Simplify. 36n2 • 4n3
Use the Commutative Property. 36 • 4 • n2 • n3
Multiply the constants and add the exponents. 144n5

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 10.47:

Simplify: (7n)2 (2n12).

Exercise 10.48:

Simplify: (4m)2(3m3).

Example 10.25:

Simplify: (3p2q)4(2pq2)3.

Solution
Use the Power of a Product Property. 34(p2)4q4 • 23p3(q2)3
Use the Power Property. 81p8q4 • 8p3q6
Use the Commutative Property. 81 • 8 • p8 • p3 • q4 • q6
Multiply the constants and add the exponents for each variable. 648p11q10

Exercise 10.49:

Simplify: (u3v2)5(4uv4)3.

Exercise 10.50:

Simplify: (5x2y3)2(3xy4)3.

Multiply Monomials

Since a monomial is an algebraic expression, we can use the properties for simplifying expressions with exponents to multiply the monomials.

Example 10.26:

Multiply: (4x2)(−5x3).

Solution
Use the Commutative Property to rearrange the factors. 4 • (−5) • x2 • x3
Multiply. −20x5

Exercise 10.51:

Multiply: (7x7)(−8x4).

Exercise 10.52:

Multiply: (−9y4)(−6y5).

Example 10.27:

Multiply: \(\left(\dfrac{3}{4} c^{3} d\right)\)(12cd2).

Solution
Use the Commutative Property to rearrange the factors. \(\frac{3}{4}\) • 12 • c3 • c • d • d2
Multiply. 9c4d3

Exercise 10.53:

Multiply: \(\left(\dfrac{4}{5} m^{4} n^{3} d\right)\)(15mn3).

Exercise 10.54:

Multiply: \(\left(\dfrac{2}{3} p^{5} q d\right)\)(18p6q7).

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Exponent Properties 

Exponent Properties 2

Practice Makes Perfect

Simplify Expressions with Exponents

In the following exercises, simplify each expression with exponents.

  1. 45
  2. 103
  3. \(\left(\dfrac{1}{2}\right)^{2}\) 
  4. \(\left(\dfrac{3}{5}\right)^{2}\) 
  5. (0.2)3
  6. (0.4)3
  7. (−5)4
  8. (−3)5
  9. −54
  10. −35
  11. −104
  12. −26
  13. \(- \left(\dfrac{2}{3}\right)^{3}\) 
  14. \(- \left(\dfrac{1}{4}\right)^{4}\) 
  15. −0.52
  16. −0.14

Simplify Expressions Using the Product Property of Exponents

In the following exercises, simplify each expression using the Product Property of Exponents.

  1. x3 • x6
  2. m4 • m2
  3. a • a4
  4. y12 • y
  5. 35 • 39
  6. 510 • 56
  7. z • z2 • z3
  8. a • a3 • a5
  9. xa • x2
  10. yp • y3
  11. ya • yb
  12. xp • xq

Simplify Expressions Using the Power Property of Exponents

In the following exercises, simplify each expression using the Power Property of Exponents.

  1. (u4)2
  2. (x2)7
  3. (y5)4
  4. (a3)2
  5. (102)6
  6. (28)3
  7. (x15)6
  8. (y12)8
  9. (x2)y
  10. (y3)x
  11. (5x)y
  12. (7a)b

Simplify Expressions Using the Product to a Power Property

In the following exercises, simplify each expression using the Product to a Power Property.

  1. (5a)2
  2. (7x)2
  3. (−6m)3
  4. (−9n)3
  5. (4rs)2
  6. (5ab)3
  7. (4xyz)4
  8. (−5abc)3

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

  1. (x2)4 • (x3)2
  2. (y4)3 • (y5)2
  3. (a2)6 • (a3)8
  4. (b7)5 • (b2)6
  5. (3x)2(5x)
  6. (2y)3 (6y)
  7. (5a)2(2a)3
  8. (4b)2(3b)3
  9. (2m6)3
  10. (3y2)4
  11. (10x2y)3
  12. (2mn4)5
  13. (−2a3b2)4
  14. (−10u2v4)3
  15. \(\left(\dfrac{2}{3} x^{2} y \right)^{3}\) 
  16. \(\left(\dfrac{7}{9} p q^{4} \right)^{2}\) 
  17. (8a3)2(2a)4
  18. (5r2)3(3r)2
  19. (10p4)3(5p6)2
  20. (4x3)3(2x5)4
  21. \(\left(\dfrac{1}{2} x^{2} y^{3} \right)^{4}\) (4x5y3)2
  22. \(\left(\dfrac{1}{3} m^{3} n^{2} \right)^{4}\) (9m8n3)2
  23. (3m2n)2(2mn5)4
  24. (2pq4)3(5p6q)2

Multiply Monomials

In the following exercises, multiply the following monomials.

  1. (12x2)(−5x4)
  2. (−10y3)(7y2)
  3. (−8u6)(−9u)
  4. (−6c4)(−12c)
  5. \(\left(\dfrac{1}{5} r^{8} \right)\) (20r3)
  6. \(\left(\dfrac{1}{4} a^{5} \right)\) (36a2)
  7. (4a3b)(9a2b6)
  8. (6m4n3)(7mn5)
  9. \(\left(\dfrac{4}{7} x y^{2} \right)\) (14xy3)
  10. \(\left(\dfrac{5}{8} u^{3} v \right)^{3}\) (24u5v)
  11. \(\left(\dfrac{2}{3} x^{2} y \right) \left(\dfrac{3}{4} x y^{2} \right)\) 
  12. \(\left(\dfrac{3}{5} m^{3} n^{2} \right) \left(\dfrac{5}{9} m^{2} n^{3} \right)\) 

Everyday Math

  1. 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 62, on the third round is 63, 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 62
3 63
8 ?
  1. 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

  1. Use the Product Property for Exponents to explain why x • x = x2.
  2. Explain why −53 = (−5)3 but −54 ≠ (−5)4.
  3. Jorge thinks \(\left(\dfrac{1}{2}\right)^{2}\) is 1. What is wrong with his reasoning?
  4. Explain why x• x5 is x8, and not x15.

Self Check

(a) After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

CNX_BMath_Figure_AppB_060.jpg

(b) After reviewing this checklist, what will you do to become confident for all objectives?

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