3.1 Simplifying Monomials Using Laws of Exponentsname

3.1 Simplifying Monomials using Laws of ExponentsName:

● Recall – Exponential Expressions

bn

Expand and simplify each expression.

a. 22 • 23b. (x3)2 c.

● The Properties of exponents apply onlyto monomials.

Monomials / Binomials / Trinomials
Examples:

● The Properties of Exponents

Property / Operation(s) / Examples
Zero Exponent Property
b0 = 1 / none / 20 = 1 = 1 0.25x0 = 0.25
Exponent Property of 1
b1 = b or b = b1 / none / ‒21 = ‒2 = (5x2y3)1 = 5x2y3
Negative Exponent
b‒n = = or = = bn / Find the Reciprocal.
Remove the negative sign. / = = = =72 = 49
Product
bn • bm = bm + n or (bn)(bm) = bm + n / Multiplythe coefficients.
Add the exponents. / 22(24) = 26 = 64 ‒x2• ‒5x = ‒1(‒5)x2+1
= 5x3
Quotient
= bn – m / Dividethe coefficients.
Subtract the exponents. / = y7 ‒ 4 = y3 = =
Power-to-a-Power
(bm)n = bm • n
(apbm)n = ap • n bm • n
= = / Distribute the exponent to each base’s exponent.
Multiply the exponents. / (72)3 = 72 •3 = 76 = 117,649
(x2)4 = x2 •4 = x8
(‒2a2b)3 = (‒21a2b1)3 = ‒21 •3a2 •3b1 •3
= ‒23a6b4 = ‒8a6b4

● Negative Exponents – Find thereciprocalandremovethe negative sign.

Example 1: Simplify each expression.

a. 4‒3b. x‒9

c. -3a‒2bd. e. f. 5x2y‒2

● Multiplying Powers with the same Base– multiplythe coefficients, addthe exponents.

Example 2: Simplify each expression.

a. 64 • 6‒2b. y4 • y2

c. 2x2 • xd. (‒3a4)(3a2)e. (x‒5)(‒3x7)

● Dividing Powers with the same Base – dividethe coefficients, subtractthe exponents

Example 3: Simplify each expression.

a. b.

c. d. e.

● Raising a Power to a Power – distributethe exponent to each base’s exponent andmultiplythe exponents.

Example 4: Simplify each expression.

a. (‒34)3 b. (x2)4c. (‒2x)2

d. (y2)‒5` e. (‒5xy2)3 f. 2x3(4x)2g.

Practice:

Simplify each expression.Example 1

1. 1302. 5‒33.  (7)‒24. 46‒15. 60

6. 12x‒27.6bc08.  11x09. 3m‒8p010.8‒2q3r‒5

11. 10a 4b012. 13. 15. 16.

Simplify each expression. Example 2

17. z8z518. 4k‒3· 6k419.(5b3)(3b6)20. (13x‒8)(3x10)

21. (2h5)(4h‒3)22.8n · 11n923.(t3)(t6)(t9)24. (x–8)(4x12)

25. (5d‒5)(6d2)26.mn2·m2n‒4·mn‒127. (6a3b‒2)( 4ab‒8)

28. (12mn)( m3n‒2p5)(2m)29. q4·r‒5·q3·r530. 3c7d‒2· 5c‒3d

Simplify each expression.Example 3

31. 32. 33. 34.

35. 36. 37. 38.

39. 40. 41. 42.

Simplify each expression. Example 4

43. (z5)344. (m4)1045. (v7)246. (k4)3

47. (x7)‒248. (r4)‒649.b(b‒8)‒350. h2(h7)0

51. (m2)7n552. (x6)2(y3)053. (g5)‒5(g6)‒254. (v2)3(w4)‒3

55. (4a3)2a556. (m4n3)7(m4)357. (xy2)(xy2)‒1 58. z(y‒5z7)‒1y‒5

Mixed practice:

59. 60. 61. 62.

63. 64. 65. 66.

67. 68. 69. 70.