m81337 Divide Monomials By the end of this section, you will be able to: Simplify expressions using the Quotient Property of Exponents Simplify expressions with zero exponents Simplify expressions using the Quotient to a Power Property Simplify expressions by applying several properties Divide monomials f1ad5ff7-3a6e-415c-83c7-abf85ee94df7 Before you get started, take this readiness quiz. Simplify: 824.If you missed the problem, review . 13 Simplify: (2m3)5.If you missed the problem, review . 32m15 Simplify: 12x12y.If you missed the problem, review . xy

Earlier in this chapter, we developed the properties of exponents for multiplication. We summarize these properties here. 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(ab)m=ambm Now we will look at the exponent properties for division. A quick memory refresher may help before we get started. In Fractions you learned that fractions may be simplified by dividing out common factors from the numerator and denominator using the Equivalent Fractions Property. This property will also help us work with algebraic fractions—which are also quotients. If a,b,c are whole numbers where b≠0,c≠0, then ab=a·cb·canda·cb·c=ab As before, we'll try to discover a property by looking at some examples. Considerx5x2andx2x3What do they mean?x⋅x⋅x⋅x⋅xx⋅xx⋅xx⋅x⋅xUse the Equivalent Fractions Property.x⋅x⋅x⋅x⋅xx⋅x⋅1x⋅x⋅1x⋅x⋅xSimplify.x31xNotice that in each case the bases were the same and we subtracted the exponents. When the larger exponent was in the numerator, we were left with factors in the numerator and 1 in the denominator, which we simplified. When the larger exponent was in the denominator, we were left with factors in the denominator, and 1 in the numerator, which could not be simplified. We write: x5x2x2x3x5−21x3−2x31x If a is a real number, a≠0, and m,n are whole numbers, thenaman=am−n,m>nandaman=1an−m,n>m A couple of examples with numbers may help to verify this property. 3432=?34−25253=?153−2819=?3225125=?1519=9✓15=15✓ When we work with numbers and the exponent is less than or equal to 3, we will apply the exponent. When the exponent is greater than 3, we leave the answer in exponential form. Simplify: ⓐ x10x8ⓑ 2922 To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator. ⓐ Since 10 > 8, there are more factors of x in the numerator. x10x8 Use the quotient property with m>n,aman=am−n. Simplify. x2 ⓑ Since 9 > 2, there are more factors of 2 in the numerator. 2922 Use the quotient property with m>n,aman=am−n. Simplify. 27Notice that when the larger exponent is in the numerator, we are left with factors in the numerator. Simplify: ⓐ x12x9ⓑ 71475 ⓐ x³ⓑ 7⁹ Simplify: ⓐ y23y17ⓑ 81587 ⓐ y⁶ⓑ 8⁸ Simplify: ⓐ b10b15ⓑ 3335 To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator. ⓐ Since 15 > 10, there are more factors of b in the denominator. b10b15 Use the quotient property with n>m,aman=1an−m. Simplify. 1b5 ⓑ Since 5 > 3, there are more factors of 3 in the denominator. 3335 Use the quotient property with n>m,aman=1an−m. Simplify. 132 Apply the exponent. 19 Notice that when the larger exponent is in the denominator, we are left with factors in the denominator and 1 in the numerator. Simplify: ⓐ x8x15ⓑ 12111221 ⓐ 1x7ⓑ 11210 Simplify: ⓐ m17m26ⓑ 78714 ⓐ 1m9ⓑ 176 Simplify: ⓐ a5a9ⓑ x11x7 ⓐ Since 9 > 5, there are more a's in the denominator and so we will end up with factors in the denominator. a5a9 Use the Quotient Property for n>m,aman=1an−m. Simplify. 1a4 ⓑ Notice there are more factors of x in the numerator, since 11 > 7. So we will end up with factors in the numerator. x11x7 Use the Quotient Property for m>n,aman=an−m. Simplify. x4 Simplify: ⓐ b19b11ⓑ z5z11 ⓐ b8ⓑ 1z6 Simplify: ⓐ p9p17ⓑ w13w9 ⓐ 1p8ⓑ w4

A special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as an expression like amam. From earlier work with fractions, we know that 22=11717=1−43−43=1 In words, a number divided by itself is 1. So xx=1, for any x (x≠0), since any number divided by itself is 1. The Quotient Property of Exponents shows us how to simplify aman when m>n and when n<m by subtracting exponents. What if m=n? Now we will simplify amam in two ways to lead us to the definition of the zero exponent.Consider first 88, which we know is 1. 88=1 Write 8 as 23. 2323=1 Subtract exponents. 23−3=1 Simplify. 20=1 We see aman simplifies to a a0 and to 1. So a0=1. If a is a non-zero number, then a0=1. Any nonzero number raised to the zero power is 1. In this text, we assume any variable that we raise to the zero power is not zero. Simplify: ⓐ 120ⓑ y0 The definition says any non-zero number raised to the zero power is 1. ⓐ 120 Use the definition of the zero exponent. 1 ⓑ y0 Use the definition of the zero exponent. 1 Simplify: ⓐ 170ⓑ m0 ⓐ 1ⓑ 1 Simplify: ⓐ k0ⓑ 290 ⓐ 1ⓑ 1 Now that we have defined the zero exponent, we can expand all the Properties of Exponents to include whole number exponents. What about raising an expression to the zero power? Let's look at (2x)0. We can use the product to a power rule to rewrite this expression. (2x)0 Use the Product to a Power Rule. 20x0 Use the Zero Exponent Property. 1⋅1 Simplify. 1 This tells us that any non-zero expression raised to the zero power is one. Simplify: (7z)0. (7z)0 Use the definition of the zero exponent. 1 Simplify: (−4y)0. 1 Simplify: (23x)0. 1 Simplify: ⓐ (−3x2y)0ⓑ −3x2y0 ⓐ The product is raised to the zero power. (−3x2y)0 Use the definition of the zero exponent. 1 ⓑ Notice that only the variable y is being raised to the zero power. −3x2y0 Use the definition of the zero exponent. −3x2⋅1 Simplify. −3x2 Simplify: ⓐ (7x2y)0ⓑ 7x2y0 ⓐ 1ⓑ 7x² Simplify: ⓐ −23x2y0ⓑ (−23x2y)0 ⓐ −23x²ⓑ 1

Now we will look at an example that will lead us to the Quotient to a Power Property. (xy)3 This means xy⋅xy⋅xy Multiply the fractions. x⋅x⋅xy⋅y⋅y Write with exponents. x3y3 Notice that the exponent applies to both the numerator and the denominator. We see that (xy)3 is x3y3. We write:(xy)3x3y3 This leads to the Quotient to a Power Property for Exponents. If a and b are real numbers, b≠0, and m is a counting number, then(ab)m=ambm To raise a fraction to a power, raise the numerator and denominator to that power. An example with numbers may help you understand this property: (23)3=?233323⋅23⋅23=?827827=827✓ Simplify: ⓐ (58)2ⓑ (x3)4ⓒ (ym)3 ⓐ Use the Quotient to a Power Property, (ab)m=ambm. Simplify. ⓑ Use the Quotient to a Power Property, (ab)m=ambm. Simplify. ⓒ Raise the numerator and denominator to the third power. Simplify: ⓐ (79)2ⓑ (y8)3ⓒ (pq)6 ⓐ 4981ⓑ y3512ⓒ p6q6 Simplify: ⓐ (18)2ⓑ (−5m)3ⓒ (rs)4 ⓐ 164ⓑ −125m3ⓒ r4s4

We'll now summarize all the properties of exponents so they are all together to refer to as we simplify expressions using several properties. Notice that they are now defined for whole number exponents. 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=ambmQuotient Propertyaman=am−n,a≠0,m>naman=1an−m,a≠0,n>mZero Exponent Definitiona0=1,a≠0Quotient to a Power Property(ab)m=ambm,b≠0 Simplify: (x2)3x5. (x2)3x5 Multiply the exponents in the numerator, using the Power Property. x6x5 Subtract the exponents. x Simplify: (a4)5a9. a¹¹ Simplify: (b5)6b11. b¹⁹ Simplify: m8(m2)4. m8(m2)4 Multiply the exponents in the numerator, using the Power Property. m8m8 Subtract the exponents. m0 Zero power property 1 Simplify: k11(k3)3. k² Simplify: d23(d4)6. 1d Simplify: (x7x3)2. (x7x3)2 Remember parentheses come before exponents, and the bases are the same so we can simplify inside the parentheses. Subtract the exponents. (x7−3)2 Simplify. (x4)2 Multiply the exponents. x8 Simplify: (f14f8)2. f¹² Simplify: (b6b11)2. 1b10 Simplify: (p2q5)3. Here we cannot simplify inside the parentheses first, since the bases are not the same. (p2q5)3 Raise the numerator and denominator to the third power using the Quotient to a Power Property, (ab)m=ambm (p2)3(q5)3 Use the Power Property, (am)n=am⋅n. p6q15 Simplify: (m3n8)5. m15n40 Simplify: (t10u7)2. t20u14 Simplify: (2x33y)4. (2x33y)4 Raise the numerator and denominator to the fourth power using the Quotient to a Power Property. (2x3)4(3y)4 Raise each factor to the fourth power, using the Power to a Power Property. 24(x3)434y4 Use the Power Property and simplify. 16x1281y4 Simplify: (5b9c3)2. 25b281c6 Simplify: (4p47q5)3. 64p12343q15 Simplify: (y2)3(y2)4(y5)4. (y2)3(y2)4(y5)4 Use the Power Property. (y6)(y8)y20 Add the exponents in the numerator, using the Product Property. y14y20 Use the Quotient Property. 1y6 Simplify: (y4)4(y3)5(y7)6. 1y11 Simplify: (3x4)2(x3)4(x5)3. 9x⁵

We have now seen all the properties of exponents. We'll use them to divide monomials. Later, you'll use them to divide polynomials. Find the quotient: 56x5÷7x2. 56x5÷7x2 Rewrite as a fraction. 56x57x2 Use fraction multiplication to separate the number part from the variable part. 567⋅x5x2 Use the Quotient Property. 8x3 Find the quotient: 63x8÷9x4. 7x⁴ Find the quotient: 96y11÷6y8. 16y³ When we divide monomials with more than one variable, we write one fraction for each variable. Find the quotient: 42x2y3−7xy5. 42x2y3−7xy5 Use fraction multiplication. 42−7⋅x2x⋅y3y5 Simplify and use the Quotient Property. −6⋅x⋅1y2 Multiply. −6xy2 Find the quotient: −84x8y37x10y2. −12yx2 Find the quotient: −72a4b5−8a9b5. 9a5 Find the quotient: 24a5b348ab4. 24a5b348ab4 Use fraction multiplication. 2448⋅a5a⋅b3b4 Simplify and use the Quotient Property. 12⋅a4⋅1b Multiply. a42b Find the quotient: 16a7b624ab8. 2a63b2 Find the quotient: 27p4q7−45p12q. −3q65p8 Once you become familiar with the process and have practiced it step by step several times, you may be able to simplify a fraction in one step. Find the quotient: 14x7y1221x11y6. 14x7y1221x11y6 Simplify and use the Quotient Property. 2y63x4 Be very careful to simplify 1421 by dividing out a common factor, and to simplify the variables by subtracting their exponents. Find the quotient: 28x5y1449x9y12. 4y27x4 Find the quotient: 30m5n1148m10n14. 58m5n3 In all examples so far, there was no work to do in the numerator or denominator before simplifying the fraction. In the next example, we'll first find the product of two monomials in the numerator before we simplify the fraction. Find the quotient: (3x3y2)(10x2y3)6x4y5. Remember, the fraction bar is a grouping symbol. We will simplify the numerator first. (3x3y2)(10x2y3)6x4y5 Simplify the numerator. 30x5y56x4y5 Simplify, using the Quotient Rule. 5x Find the quotient: (3x4y5)(8x2y5)12x5y8. 2xy² Find the quotient: (−6a6b9)(−8a5b8)−12a10b12. −4ab⁵ Simplify a Quotient Zero Exponent Quotient Rule Polynomial Division Polynomial Division 2

Equivalent Fractions Property If a,b,c are whole numbers where b≠0,c≠0, then ab=a·cb·canda·cb·c=ab Zero Exponent If a is a non-zero number, then a0=1. Any nonzero number raised to the zero power is 1. Quotient Property for Exponents If a is a real number, a≠0, and m,n are whole numbers, then aman=am−n,m>nandaman=1an−m,n>m Quotient to a Power Property for Exponents If a and b are real numbers, b≠0, and m is a counting number, then (ab)m=ambm To raise a fraction to a power, raise the numerator and denominator to that power.

Simplify Expressions Using the Quotient Property of Exponents In the following exercises, simplify. 4842 4⁶ 31234 x12x3 x⁹ u9u3 r5r r⁴ y4y y4y20 1y16 x10x30 1031015 11012 r2r8 aa9 1a8 225 Simplify Expressions with Zero Exponents In the following exercises, simplify. 50 1 100 a0 1 x0 −70 −1 −40 ⓐ (10p)0ⓑ 10p0 ⓐ 1ⓑ 10 ⓐ (3a)0ⓑ 3a0 ⓐ (−27x5y)0ⓑ −27x5y0 ⓐ 1ⓑ −27x⁵ ⓐ (−92y8z)0ⓑ −92y8z0 ⓐ 150ⓑ 151 ⓐ 1ⓑ 15 ⓐ −60ⓑ −61 2·x0+5·y0 7 8·m0−4·n0 Simplify Expressions Using the Quotient to a Power Property In the following exercises, simplify. (32)5 24332 (45)3 (m6)3 m3216 (p2)5 (xy)10 x10y10 (ab)8 (a3b)2 a29b2 (2xy)4 Simplify Expressions by Applying Several Properties In the following exercises, simplify. (x2)4x5 x³ (y4)3y7 (u3)4u10 u² (y2)5y6 y8(y5)2 1y2 p11(p5)3 r5r4·r 1 a3·a4a7 (x2x8)3 1x18 (uu10)2 (a4·a6a3)2 a¹⁴ (x3·x8x4)3 (y3)5(y4)3 y³ (z6)2(z2)4 (x3)6(x4)7 1x10 (x4)8(x5)7 (2r35s)4 16r12625s4 (3m24n)3 (3y2·y5y15·y8)0 1 (15z4·z90.3z2)0 (r2)5(r4)2(r3)7 1r3 (p4)2(p3)5(p2)9 (3x4)3(2x3)2(6x5)2 3x⁸ (−2y3)4(3y4)2(−6y3)2 Divide Monomials In the following exercises, divide the monomials. 48b8÷6b2 8b⁶ 42a14÷6a2 36x3÷(−2x9) −18x6 20u8÷(−4u6) 18x39x2 2x 36y94y7 −35x7−42x13 56x6 18x5−27x9 18r5s3r3s9 6r2s8 24p7q6p2q5 8mn1064mn4 n68 10a4b50a2b6 −12x4y915x6y3 −4y65x2 48x11y9z336x6y8z5 64x5y9z748x7y12z6 4z3x2y3 (10u2v)(4u3v6)5u9v2 (6m2n)(5m4n3)3m10n2 10n2m4 (6a4b3)(4ab5)(12a8b)(a3b) (4u5v4)(15u8v)(12u3v)(u6v) 5u⁴v³

ⓐ 24a5+2a5ⓑ 24a5−2a5ⓒ 24a5⋅2a5ⓓ 24a5÷2a5 ⓐ 15n10+3n10ⓑ 15n10−3n10ⓒ 15n10⋅3n10ⓓ 15n10÷3n10 ⓐ 18n10 ⓑ 12n10 ⓒ 45n20 ⓓ 5 ⓐ p4⋅p6 ⓑ (p4)6 ⓐ q5⋅q3 ⓑ (q5)3 ⓐ q8 ⓑ q15 ⓐ y3y ⓑ yy3 ⓐ z6z5 ⓑ z5z6 ⓐ zⓑ 1z (8x5)(9x)÷6x3 (4y)(12y7)÷8y2 6y6 27a73a3+54a99a5 32c114c5+42c96c3 15c6 32y58y2−60y105y7 48x66x4−35x97x7 3x2 63r6s39r4s2−72r2s26s 56y4z57y3z3−45y2z25y −yz2

Memory One megabyte is approximately 106 bytes. One gigabyte is approximately 109 bytes. How many megabytes are in one gigabyte? Memory One megabyte is approximately 106 bytes. One terabyte is approximately 1012 bytes. How many megabytes are in one terabyte? 1,000,000

Vic thinks the quotient x20x4 simplifies to x5. What is wrong with his reasoning? Mai simplifies the quotient y3y by writing y3y=3. What is wrong with her reasoning? Answers will vary. When Dimple simplified −30 and (−3)0 she got the same answer. Explain how using the Order of Operations correctly gives different answers. Roxie thinks n0 simplifies to 0. What would you say to convince Roxie she is wrong? Answers will vary.

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

ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

zero exponent If a is a non-zero number, then a0=1. Any nonzero number raised to the zero power is 1.