Topic 6.2.1

1. Rules of Exponents Topic 6.2.1

2. Lesson 1.1.1 Topic 6.2.1 Rules of Exponents California Standards: 2.0 Students understand and use such operations as taking the opposite,finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents. 10.0 Studentsadd, subtract,multiply, and dividemonomials and polynomials.Students solve multistep problems, including word problems, by using these techniques. What it means for you: You’ll multiply and divide algebraic expressions using the rules of exponents. • Key words: • exponent

3. Lesson 1.1.1 Topic 6.2.1 Rules of Exponents You learned about the rules of exponents in Topic 1.3.1. In this Topic, you’ll apply those same rules to monomials and polynomials. We’ll start with a quick recap of the rules of exponentsto make sure you remember them all.

4. 1 xa Lesson 1.1.1 Topic 6.2.1 Rules of Exponents Use the Rules of Exponents to Simplify Expressions These are the same rules you learned in Chapter 1, but this time you’ll use them to simplify algebraic expressions: Rules of Exponents 1) xa·xb = xa+b 2) xa÷ xb= xa–b (if x¹ 0) 3) (xa)b = xab 4) (cx)b = cbxb 5) x0 = 1 6) x–a = (if x¹ 0) 7)

5. Rule 1) xa·xb = xa+b Topic 6.2.1 Rules of Exponents Example 1 Simplify the expression (–2x2m)(–3x3m3). Solution (–2x2m)(–3x3m3)= (–2)(–3)(x2)(x3)(m)(m3) = 6x2+3·m1+3 =6x5m4 Put all like variables together Use Rule 1 and add the powers Solution follows…

6. Rule 3) (xa)b = xabRule 4) (cx)b = cbxb Topic 6.2.1 Rules of Exponents Example 2 Simplify the expression (3a2xb3)2. Solution (3a2xb3)2= 32·a2·2·x2·b3·2 = 9a4x2b6 Use Rules 3 and 4 Solution follows…

7. Simplify the expression . Rule 2) xa÷ xb= xa–b (if x¹ 0) = = Topic 6.2.1 Rules of Exponents Example 3 Solution Rule 5) x0 = 1 Separate the expression into parts that have only one variable Use Rule 2 and subtract the powers = 2xm2 From Rule 5, anything to the power 0 is 1 Solution follows…

8. Lesson 1.1.1 Topic 6.2.1 Rules of Exponents Guided Practice Simplify each expression. 1. –3at(4a2t3) 2. (–5x3yt2)(–2x2y3t) 3. (–2x2y3)34. –2mx(3m2x – 4m2x+m3x3) 5. (–3x2t)3(–2x3t2)26. –2mc(–3m2c3 + 5mc) (–3 • 4)(a • a2)(t • t3)= –12a(1 + 2)t(1 + 3) (Rule 1)= –12a3t4 (–5 • –2)(x3 • x2)(y • y3)(t2 • t)= 10x(3 + 2)y(1 + 3)t(2 + 1) (Rule 1)= 10x5y4t3 (–2)(1 • 3)x(2 • 3)y(3 • 3) (Rule 3)= (–2)3x6y9 = –8x6y9 –2mx(–m2x + m3x3)= 2m(1 + 2)x(1 + 1) – 2m(1 + 3)x(1 + 3) (Rule 1)= 2m3x2 – 2m4x4 ((–3)3x(2 • 3)t3)((–2)2x(3 • 2)t(2 • 2)) (Rule 3)= (–27x6t3)(4x6t4) = –108x(6 + 6)t(3 + 4) (Rule 1)= –108x12t7 6m(1 + 2)c(1 + 3) – 10m(1 + 1)c(1 + 1) (Rule 1)= 6m3c4 – 10m2c2 Solution follows…

9. = (14 ÷ 4)a(2 – 7)b(4 – 4)c(8 – 0) (Rule 2) = a–5c8 (Rule 5) = (Rule 6) = (12 ÷ 8)j(8 – 2)k(–8 – –10)m(–1 – 4) (Rule 2) = j6k2m–5 = (Rule 6) = (16 ÷ 32)b(9 – 5 • 2)a(4– 3 • 2)c(–1 • 2)j4 (Rule 2) = b–1a–2c–2j4 = (Rule 6) Lesson 1.1.1 Topic 6.2.1 Rules of Exponents Guided Practice Simplify each expression. 7. 8. 9.10. = 5m(3 – 2)n(8 – 3)z(6 – 1) (Rule 2)= 5mn5z5 Solution follows…

10. Topic 6.2.1 Rules of Exponents Independent Practice Simplify. 1. 2. 3. 4a2(a2 – b2) 4. 4m2x2(x2 + x + 1) 5.a(a + 4) + 4(a + 4) 6. 2a(a – 4) – 3(a – 4) 7. m2n3(mx2 + 3nx + 2) – 4m2n3 8. 4m2n2(m3n8 + 4) – 3m3n10(m2 + 2n3) 1 4a4 – 4a2b2 4m2x4 + 4m2x3 + 4m2x2 a2 + 8a + 16 2a2 – 11a + 12 m3n3x2 + 3m2n4x – 2m2n3 m5n10 + 16m2n2 – 6m3n13 Solution follows…

11. Topic 6.2.1 Rules of Exponents Independent Practice Simplify. 9.10. 11.12. Solution follows…

12. Topic 6.2.1 Rules of Exponents Independent Practice Find the value of ? that makes these statements true. 13.m?(m4 + 2m3) = m6 + 2m5 14. m4a6(3m?a8 + 4m2a?) = 3m7a14 + 4m6a9 ? = 2 ? = 3 15.16. ? = 7 ? = 4 Solution follows…

13. Topic 6.2.1 Rules of Exponents Round Up You can apply the rules of exponents to any algebraic values. In this Topic you just dealt with monomials, but the rules work with expressions with more than one term too.