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Objectives

8.1 Laws of Exponents: Multiplying Monomials. Objectives. Define exponents and powers. Find products of powers. Simplify products of monomials. NCSCOS. 1.01, 1.02, 4.04. x m = x  x  x  . . .  x. m factors. 8.1 Laws of Exponents: Multiplying Monomials. Exponents

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Objectives

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  1. 8.1 Laws of Exponents: Multiplying Monomials Objectives • Define exponents and powers. • Find products of powers. • Simplify products of monomials. NCSCOS • 1.01, 1.02, 4.04

  2. xm = xxx. . .x m factors 8.1 Laws of Exponents: Multiplying Monomials Exponents Product-of-Powers Property For all nonzero real numbers x and all integers m and n, xmxn = xm+n x3 x5 = x3+5 = x8 x3c  x4c = x7c = (x + c)3d (x + c)d  (x + c)2d

  3. 8.1 Laws of Exponents: Multiplying Monomials Text Book Example #1 Simplify. Leave in exponential form. 72 = 14 = 49 36 (35)(3)= 53 = 125 (82)(82) = 84 34 = 81 z9 (z6)(z3) = 64 26 = (az)(a3) = az+3 35 = 243 (7a)(7b) = 7a+b 27 = 128

  4. 8.1 Laws of Exponents: Multiplying Monomials Text Book Example #3 Suppose that a colony of bacteria doubles in size every hour. If the colony contains 1000 bacteria at noon, how many bacteria will the colony contain at 3 pm and at 5pm on the same day? This can be solved two ways: Noon 1pm 2pm 3 pm 4 pm 5pm 2000 1000 8000 16,000 4000 32,000 32,000 1000 • 25 = 8000 Or, 1000 • 23 = (where, 1000 is the original colony, times 2 means doubled, and raised to the power of three and five is the elapsed time in hours; 3, and 5 hours)

  5. 8.1 Laws of Exponents: Multiplying Monomials Text Book Example #4 Simplify. Hint: multiply coefficients and add exponents having the same base (5t)(–30t2) = –150t3 (–4a2b)(–ac2)(3b2c2) = 12a3b3c4 (3m2)(60mp2) = 180m3p2 (8xz)(–10y)(–2yz2) = 160xy2z3

  6. 8.1 Laws of Exponents: Multiplying Monomials Text Book Example #5 The volume, V, of a right rectangular prism can be found by using the formula V = lwh, where l is thelength, w is the width, and h is the height. Suppose that a prisim has a length of 2xy, a width of 3xy, and a height of 6xyz. Find the volume. (2xy)(3xy)(6xyz) = (36x3y3z)

  7. 8.2 Laws of Exponents: Powers and Products NCSCOS • 1.01, 1.02, 4.04 Objectives Find the Power-of-a-Power Find the Power-of-a-Product

  8. 8.2 Laws of Exponents: Powers and Products Text Book Example #1 Power-of-a-Power Property For all real numbers x and all integers m and n, (xm)n = xmn. (23)4 = 23 • 4 = 212 = 4096 (p2)5 = p10 (xm)2 = x2m (4a)2 = 16a2

  9. 8.2 Laws of Exponents: Powers and Products Text Book Example #2 Power-of-a-Product Property For all nonzero real numbers x and y and all integers n, (xy)n = xnyn. (xy2)3 = x3y6 (xy)2 = x2y2 (xy2)2 = x2y4 4r8 (r2)4 =

  10. 8.2 Laws of Exponents: Powers and Products Text Book Example #4 Simplify. x6y9 (x2y3)3 = (ab2cn)5 = a5b10c5n (ab2c3d)n = anb2nc3ndn (-5xy2z4)3 = -125x3y6z12

  11. 8.2 Laws of Exponents: Powers and Products Text Book Example #5 The volume, V, of a sphere is given by the formula Suppose that the radius, r, of a sphere is equal to 6s2, where, s, is the radius of a smaller sphere. Find the volume of the both spheres in terms of the radius. Smaller sphere: = 6s2 Larger sphere: s

  12. 8.2 Laws of Exponents: Powers and Products Text Book Example #6 Even powers on a negative # = positive #. Odd powers on a negative # = negative #. y100 (-t)4 = t4 (-y)100 = y4 (-y)4 = (-3y)4 = 81y4 -64x15 -t5 (-4x5)3 = (-t)5 = (-3h9)3 = -125x3 (-5x)3 = -27h27

  13. Guided Practice Objectives • Use the laws of exponents (products of powers, power-of-a-power and power-of-a-product) to simplify monominals.

  14. 8.1 Laws of Exponents: Multiplying Monomials Guided Practice Simplify in exponential form. Evaluate. 625 54= (25)(24 ) = 29 23 = 8 (f5)(f9) = f14 t8 (t3)(t5) = 34 = 81 11 111 = (xt)(xr) = xr+t Simplify each product. (8p3)(40m7p6) = 320m7p9 (–4x3z2)(–6y5)(-y3z7) = -24x3y8z9

  15. 8.1 Laws of Exponents: Multiplying Monomials Guided Practice Suppose that a colony of bacteria doubles in size every half-hour. The colony contains 2000 bacteria at 10 am, how many bacteria will the colony contain at 1 pm and at 2 pm? (10 – 1 pm = 3-hours and 6 ½-hrs) 128,000 10 – 1pm= 2000 • 26 = (10 – 2 pm = 4-hours and 8 ½-hrs) 10 – 2 pm = 2000 • 28 = 512,000

  16. 8.1 Laws of Exponents: Multiplying Monomials Guided Practice The volume, V, of a right rectangular prism can be found by using the formula V = lwh, where l is thelength, w is the width, and h is the height. Suppose that a prisim has a length of 3rt, a width of 5rt, and a height of 2rty. Find the volume. (3rt)(5rt)(2rty) = (30r3t3y)

  17. 8.2 Laws of Exponents: Powers and Products Guided Practice Simplify and find the value when possible. (35)2 = 310 = 59,049 (b4)9 = b36 (qrs)t = qtrtst (z7)y = z7y x4my8z4 (xmy2z)4 = (ab)8 = (-6)3 = a8b8 -216 b10 (-b)10 = (x2y4)3 = x6y12 -125k3 (-5k)3 = (c2h7)4 = c8h28 -1 m6n15 (m2n5)3 = (-1)1001 =

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