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Lesson 13, Section 2Three Rules of ExponentsRule 1. Same base aman = am + n "To multiply powers of the same base, add the exponents." For example, a2a3 = a5. Why do we add the exponents? Because of what the symbols mean. Section 1. Example 1. Multiply 3x2 · 4x5 · 2x Solution. The problem means (Lesson 5): Multiply the numbers, then combine the powers of x : 3x2 · 4x5 · 2x = 24x8 Two factors of x -- x2 -- times five factors of x -- x5 -- times one factor of x, produce a total of 2 + 5 + 1 = 8 factors of x : x8. Problem 1. Multiply. Apply the rule Same Base. To see the answer, pass your mouse over the colored area.
Problem 2. Distinguish the following: x · x and x + x. x · x = x². x + x = 2x. Example 2. Compare the following: a) x · x5 b) 2 · 25 Solution. a) x · x5 = x6 b) 2 · 25 = 26 Part b) has the same form as part a). It is part a) with x = 2. One factor of 2 multiplies five factors of 2 producing six factors of 2. 2 · 2 = 4 is not correct here. Problem 3. Apply the rule Same Base.
Problem 4. Apply the rule Same Base.
Rule 2: Power of a product of factors (ab)n = anbn "Raise each factor to that same power." For example, (ab)3 = a3b3. Why may we do that? Again, according to what the symbols mean: (ab)3 = ab · ab · ab = aaabbb = a3b3. The order of the factors does not matter: ab · ab · ab = aaabbb. Problem 5. Apply the rules of exponents.
Rule 3: Power of a power (am)n = amn "To take a power of a power, multiply the exponents." For example, (a2)3 = a2 · 3 = a6. Why do we do that? Again, because of what the symbols mean: (a2)3 = a2a2a2 = a3 · 2 = a6 Problem 6. Apply the rules of exponents.
Example 3. Apply the rules of exponents: (2x3y4)5 Solution. Within the parentheses there are three factors: 2, x3, and y4. According to Rule 2 we must take the fifth power of each one. But to take a power of a power, we multiply the exponents. Therefore, (2x3y4)5 = 25x15y20 Problem 7. Apply the rules of exponents.
Problem 8. Apply the rules of exponents. a) 2x5y4(2x3y6)5 = 2x5y4 · 25x15y30 = 26x20y34 b) abc9(a2b3c4)8 = abc9 · a16b24c32 = a17b25c41 Problem 9. Use the rules of exponents to calculate the following. a) (2 · 10)4 = 24 · 104 = 16 · 10,000 = 160,000 b) (4 · 102)3 = 43 · 106 = 64,000,000 c) (9 · 104)2 = 81 · 108 = 8,100,000,000 The powers of 10 have as many 0's as the exponent of 10. Example 4. Square x4. Solution. (x4)2 = x8. To square a power, double the exponent. Problem 10. Square the following.
Part c) illstrates: The square of a number is never negative. (−6)(−6) = +36. The Rule of Signs. Problem 11. Apply a rule of exponents -- if possible.
In summary: Add the exponents when the same base appears twice: x2x4 = x6. Multiply the exponents when the base appears once -- and in parentheses: (x2)5 = x10. Problem 12. Apply the rules of exponents.
Problem 13. Apply a rule of exponents or add like terms -- if possible. a) 2x2 + 3x4 Not possible. These are not like terms. b) 2x2 · 3x4 = 6x6. Rule 1. c) 2x3 + 3x3 = 5x3. Like terms. The exponent does not change. d) x2 + y2 Not possible. These are not like terms. e) x2 + x2 = 2x2. Like terms. f) x2 · x2 = x4. Rule 1 g) x2 · y3 Not possible. Different bases. h) 2 · 26 = 27. Rule 1 i) 35 + 35 + 35 = 3 · 35 (On adding those like terms) = 36. We will continue the rules of exponents in Lesson 21. Next Lesson: Multiplying out. The distributive rule. Copyright © 2021 Lawrence Spector Questions or comments? E-mail: teacher@themathpage.com |