m81335 Use Multiplication Properties of Exponents By the end of this section, you will be able to: Simplify expressions with exponents Simplify expressions using the Product Property of Exponents Simplify expressions using the Power Property of Exponents Simplify expressions using the Product to a Power Property Simplify expressions by applying several properties Multiply monomials 1ed2d0c6-4b2c-496f-8e97-236eab9dafe0 Before you get started, take this readiness quiz. Simplify: 34·34. If you missed the problem, review . 916 Simplify: (−2)(−2)(−2). If you missed the problem, review . −8

Remember that an exponent indicates repeated multiplication of the same quantity. For example, 24 means to multiply four factors of 2, so 24 means 2·2·2·2. This format is known as exponential notation. This is read a to the mth power. In the expression am, the exponent tells us how many times we use the base a as a factor. Before we begin working with variable expressions containing exponents, let’s simplify a few expressions involving only numbers. Simplify: ⓐ 53ⓑ 91 ⓐ 53 Multiply 3 factors of 5. 5·5·5 Simplify. 125 ⓑ 91 Multiply 1 factor of 9. 9 Simplify: ⓐ 43ⓑ 111 ⓐ 64ⓑ 11 Simplify: ⓐ 34ⓑ 211 ⓐ 81ⓐ 21 Simplify: ⓐ (78)2ⓑ (0.74)2 ⓐ (78)2 Multiply two factors. (78)(78) Simplify. 4964 ⓑ (0.74)2 Multiply two factors. (0.74)(0.74) Simplify. 0.5476 Simplify: ⓐ (58)2ⓑ (0.67)2 ⓐ 2564ⓑ 0.4489 Simplify: ⓐ (25)3ⓑ (0.127)2 ⓐ 8125ⓑ 0.016129 Simplify:ⓐ (−3)4ⓑ −34 ⓐ (−3)4 Multiply four factors of −3. (−3)(−3)(−3)(−3) Simplify. 81 ⓑ −34 Multiply two factors. −(3·3·3·3) Simplify. −81 Notice the similarities and differences in parts ⓐ and ⓑ. Why are the answers different? In part ⓐ the parentheses tell us to raise the (−3) to the 4^th power. In part ⓑ we raise only the 3 to the 4^th power and then find the opposite. Simplify: ⓐ (−2)4ⓑ −24 ⓐ 16ⓑ −16 Simplify: ⓐ (−8)2ⓑ −82 ⓐ 64ⓑ −64

You have seen that when you combine like terms by adding and subtracting, you need to have the same base with the same exponent. But when you multiply and divide, the exponents may be different, and sometimes the bases may be different, too. We’ll derive the properties of exponents by looking for patterns in several examples. All the exponent properties hold true for any real numbers, but right now we will only use whole number exponents. First, we will look at an example that leads to the Product Property. What does this mean? How many factors altogether? So, we have Notice that 5 is the sum of the exponents, 2 and 3. We write: x2⋅x3x2+3x5 The base stayed the same and we added the exponents. This leads to the Product Property for Exponents. If a is a real number and m,n are counting numbers, thenam·an=am+n To multiply with like bases, add the exponents. An example with numbers helps to verify this property. 22·23=?22+34·8=?2532=32✓ Simplify: x5·x7. x5·x7 Use the product property, am·an=am+n. Simplify. x12 Simplify: x7·x8. x¹⁵ Simplify: x5·x11. x¹⁶ Simplify: b4·b. b4·b Rewrite, b=b1. b4·b1 Use the product property, am·an=am+n. Simplify. b5 Simplify: p9·p. p¹⁰ Simplify: m·m7. m⁸ Simplify: 27·29. 27·29 Use the product property, am·an=am+n. Simplify. 216 Simplify: 6·69. 6¹⁰ Simplify: 96·99. 9¹⁵ Simplify: y17·y23. y17·y23 Notice, the bases are the same, so add the exponents. Simplify. y40 Simplify: y24·y19. y⁴³ Simplify: z15·z24. z³⁹ We can extend the Product Property of Exponents to more than two factors. Simplify: x3·x4·x2. x3·x4·x2 Add the exponents, since the bases are the same. Simplify. x9 Simplify: x7·x5·x9. x²¹ Simplify: y3·y8·y4. y¹⁵

Now let’s look at an exponential expression that contains a power raised to a power. See if you can discover a general property. What does this mean? How many factors altogether? So, we have Notice that 6 is the product of the exponents, 2 and 3. We write: (x2)3x2⋅3x6 We multiplied the exponents. This leads to the Power Property for Exponents. If a is a real number and m,n are whole numbers, then(am)n=am·n To raise a power to a power, multiply the exponents. An example with numbers helps to verify this property. (52)3=?52·3(25)3=?5615,625=15,625✓ Simplify: ⓐ (x5)7ⓐ (36)8 ⓐ (x5)7 Use the Power Property, (am)n=am·n. Simplify. x35 ⓑ (36)8 Use the Power Property, (am)n=am·n. Simplify. 348 Simplify: ⓐ (x7)4ⓑ (74)8 ⓐ x²⁸ⓑ 7³² Simplify: ⓐ (x6)9ⓑ (86)7 ⓐ x⁵⁴ⓑ 8⁴²

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)3is23·x3 We write: (2x)3 23·x3 The exponent applies to each of the factors. This leads to the Product to a Power Property for Exponents. If a and b are real numbers and m is a whole number, then(ab)m=ambm To raise a product to a power, raise each factor to that power. An example with numbers helps to verify this property: (2·3)2=?22·3262=?4·936=36✓ Simplify: (−11x)2. (−11x)2 Use the Power of a Product Property, (ab)m=ambm. Simplify. 121x2 Simplify: (−14x)2. 196x² Simplify: (−12a)2. 144a² Simplify: (3xy)3. (3xy)3 Raise each factor to the third power. Simplify. 27x3y3 Simplify: (−4xy)4. 256x⁴y⁴ Simplify: (6xy)3. 216x³y³

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. If a,b are real numbers and m,n are whole numbers, then Product Propertyam·an=am+nPower Property(am)n=am·nProduct to a Power Property(ab)m=ambm Simplify: (x2)6(x5)4. (x2)6(x5)4 Use the Power Property. x12·x20 Add the exponents. x32 Simplify: (x4)3(x7)4. x⁴⁰ Simplify: (y9)2(y8)3. y⁴² Simplify: (−7x3y4)2. (−7x3y4)2 Take each factor to the second power. (−7)2(x3)2(y4)2 Use the Power Property. 49x6y8 Simplify: (−8x4y7)3. −512x¹²y²¹ Simplify: (−3a5b6)4. 81a²⁰b²⁴ Simplify: (6n)2(4n3). (6n)2(4n3) 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. Simplify: (7n)2(2n12). 98n¹⁴ Simplify: (4m)2(3m3). 48m⁵ Simplify: (3p2q)4(2pq2)3. (3p2q)4(2pq2)3 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 Simplify: (u3v2)5(4uv4)3. 64u¹⁸v²² Simplify: (5x2y3)2(3xy4)3. 675x⁷y¹⁸

Since a monomial is an algebraic expression, we can use the properties for simplifying expressions with exponents to multiply the monomials. Multiply: (4x2)(−5x3). (4x2)(−5x3) Use the Commutative Property to rearrange the factors. 4·(−5)·x2·x3 Multiply. −20x5 Multiply: (7x7)(−8x4). −56x¹¹ Multiply: (−9y4)(−6y5). 54y⁹ Multiply: (34c3d)(12cd2). (34c3d)(12cd2) Use the Commutative Property to rearrange the factors. 34·12·c3·c·d·d2 Multiply. 9c4d3 Multiply: (45m4n3)(15mn3). 12m⁵n⁶ Multiply: (23p5q)(18p6q7). 12p¹¹q⁸ Exponent Properties Exponent Properties 2

Exponential Notation This is read a to the mth power. Product Property of Exponents If a is a real number and m,n are counting numbers, then am·an=am+n To multiply with like bases, add the exponents. Power Property for Exponents If a is a real number and m,n are counting numbers, then (am)n =am⋅n Product to a Power Property for Exponents If a and b are real numbers and m is a whole number, then(ab)m=ambm

Simplify Expressions with Exponents In the following exercises, simplify each expression with exponents. 45 1,024 103 (12)2 14 (35)2 (0.2)3 0.008 (0.4)3 (−5)4 625 (−3)5 −54 −625 −35 −104 −10,000 −26 (−23)3 −827 (−14)4 −0.52 −0.25 −0.14 Simplify Expressions Using the Product Property of Exponents In the following exercises, simplify each expression using the Product Property of Exponents. x3·x6 x⁹ m4·m2 a·a4 a⁵ y12·y 35·39 3¹⁴ 510·56 z·z2·z3 z⁶ a·a3·a5 xa·x2 x^a+2 yp·y3 ya·yb y^a+b xp·xq Simplify Expressions Using the Power Property of Exponents In the following exercises, simplify each expression using the Power Property of Exponents. (u4)2 u⁸ (x2)7 (y5)4 y²⁰ (a3)2 (102)6 10¹² (28)3 (x15)6 x⁹⁰ (y12)8 (x2)y x^2y (y3)x (5x)y 5^xy (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. (5a)2 25a² (7x)2 (−6m)3 −216m³ (−9n)3 (4rs)2 16r²s² (5ab)3 (4xyz)4 256x⁴y⁴z⁴ (−5abc)3 Simplify Expressions by Applying Several Properties In the following exercises, simplify each expression. (x2)4·(x3)2 x¹⁴ (y4)3·(y5)2 (a2)6·(a3)8 a³⁶ (b7)5·(b2)6 (3x)2(5x) 45x³ (2y)3(6y) (5a)2(2a)3 200a⁵ (4b)2(3b)3 (2m6)3 8m¹⁸ (3y2)4 (10x2y)3 1,000x⁶y³ (2mn4)5 (−2a3b2)4 16a¹²b⁸ (−10u2v4)3 (23x2y)3 827x6y3 (79pq4)2 (8a3)2(2a)4 1,024a¹⁰ (5r2)3(3r)2 (10p4)3(5p6)2 25,000p²⁴ (4x3)3(2x5)4 (12x2y3)4(4x5y3)2 x¹⁸y¹⁸ (13m3n2)4(9m8n3)2 (3m2n)2(2mn5)4 144m⁸n²² (2pq4)3(5p6q)2 Multiply Monomials In the following exercises, multiply the following monomials. (12x2)(−5x4) −60x⁶ (−10y3)(7y2) (−8u6)(−9u) 72u⁷ (−6c4)(−12c) (15r8)(20r3) 4r¹¹ (14a5)(36a2) (4a3b)(9a2b6) 36a⁵b⁷ (6m4n3)(7mn5) (47xy2)(14xy3) 8x²y⁵ (58u3v)(24u5v) (23x2y)(34xy2) 12x3y3 (35m3n2)(59m2n3)

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 ? 1,679,616 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 ?

Use the Product Property for Exponents to explain why x·x=x2. Answers will vary. Explain why −53=(−5)3 but −54≠(−5)4. Jorge thinks (12)2 is 1. What is wrong with his reasoning? Answers will vary. Explain why x3·x5 is x8, and not x15.

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

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