MODULE 3

• Writer: Alden O. Madregalejo
• Cover Illustrator: Joel J. Estudillo
• MATHEMATICS Quarter 2 – Module 3: Laws Involving Positive Integral Exponents
• Department of Education National Capital Region SCHOOLS DIVISION OFFICE MARIKINA CITY
• Exponent


  A number or letter written above and to the right of a mathematical expression called the base
• Base


  The mathematical expression that the exponent is written on
• 𝑎𝑥


  �� is the base and 𝑥 is the exponent. 𝑎 is multiplied by itself 𝑥 times.
• 𝑎−𝑥

  If 𝑎 is a positive number and 𝑥 is negative number as the exponent, then 𝑎−𝑥 becomes 1/𝑎𝑥
• 𝑎0

  If 𝑎 is a positive number and its exponent is zero (0), then 𝑎0 becomes 1
• Product Law of Exponent

  1. Multiplying powers containing the same base, simply add their exponents

  2. 𝑎𝑥 ∙ 𝑎�� = 𝑎𝑥+𝑦

  3. (𝑎/��)𝑥 ∙ (𝑎/𝑏)𝑦 = (𝑎/𝑏)𝑥+𝑦
• Product Law of Exponent


  • 𝑥2 ∙ 𝑥4 = 𝑥2+4 = 𝑥6
  • 𝑎−2 ∙ 𝑎−3 ∙ 𝑎−4 = 𝑎(−2)+(−3)+(−4) = 𝑎−9 = 1/𝑎9
  • 23 ∙ 27 = 23+7 = 210 = 1024
  • 30 ∙ 35 ∙ 3−3 = 30+5+(−3) = 32 = 9
  • (23)3 ∙ (23)−3 = (23)3+(−3) = (23)0 = 1
  • 3𝑥2 ∙ 4𝑥7 = (3 ∙ 4)(𝑥)2+7 = 12𝑥9
• Quotient Law of Exponent

  1. Dividing powers containing the same base, simply subtract their exponents

  2. 𝑎𝑥/𝑎𝑦 = ����−𝑦 if 𝑥 > 𝑦

  3. 𝑎𝑥/𝑎𝑦 = 𝑎0 = 1 if 𝑥 = 𝑦

  4. 𝑎𝑥/𝑎𝑦 = ����−𝑦 if 𝑥 < 𝑦, the resulting power is negative = 1/𝑎𝑥−𝑦
• Quotient Law of Exponent

  • 𝑥10/𝑥6 = 𝑥10−6 = 𝑥4
  • 𝑥10/𝑥10 = 𝑥10−10 = 𝑥0 = 1
  • 𝑥6/𝑥10 = 𝑥6−10 = 𝑥−4 = 1/𝑥4
  • 12𝑎2𝑏3/3𝑎2𝑏2 = 12/3 𝑎2−2𝑏3−2 = 4𝑎0𝑏1 = 4𝑏
  • 34𝑎2𝑏8/32𝑎13 = 34−2𝑎2−13𝑏8 = 32𝑎−11𝑏8 = (9)(1/𝑎11)𝑏8 = 9𝑏8/𝑎11
  • 24𝑎7𝑏−3/22𝑎5𝑏0 = 24−2𝑎7−5𝑏(−3)−0 = 22𝑎2𝑏−3 = (4)(𝑎2)(1/𝑏3) = 4𝑎2/𝑏3
• Power Law of Exponent

  1. When a power is raised to another power, simply multiply their exponents

  2. (𝑎��)𝑦 = 𝑎𝑥𝑦

  3. (𝑎/𝑏)𝑥 = 𝑎𝑥/𝑏𝑥, 𝑏 ≠ 0

  4. (𝑎𝑥/𝑏𝑦)𝑧 = 𝑎𝑥𝑧/𝑏𝑦𝑧, 𝑏 ≠ 0
• Raised to another power

  Multiply their exponents
• I. (ax)y


  axy
• II. (a/b)x

  ax/bx, b ≠ 0
• III. (ax/by)z

  axz/byz, b ≠ 0
• Apply the Power Law of Exponent
• Apply the Quotient Law of Exponent
• Simplify by multiplying (1/34)(a4)(1) then divided by x8, then multiplying (1)(a4) divided by 81
• Fill up the boxes with the laws of exponent and explain each by illustration