8th Grade - Laws of Exponents

Introduction

  • Have you seen any number with another number (positive or negative) on the top right? You might have already; those are called Exponents.
  • 23 or 2-4 are some examples of exponents, also called exponent expressions.
  • Exponents are nothing but repeated multiplication.
  • In 23, 2 is the base, and 3 is the exponent. In the case of 3x2, 3 is called the coefficient, x is the base, and 2 is the exponent.
  • 23 is read as two to the third power and also as two cubed.
  • 74 is read as seven to the fourth power.
  • Exponents can be positive or negative.

Negative Exponents

  • 3-4 is an example of negative exponents.
  • Negative exponents can be solved by finding a reciprocal (dropping that number under 1)
  • For example, 3-4 is same as 134 which is same as 181

Adding & Subtracting Exponents

  • If the base and exponents are the same, you only need to add/subtract the coefficient.
  • 2x2 + 3x2 can be added, and the result is 5x2.
  • 4x2 - 2x2 can be subtracted, and the result is 2x2.
  • 3x2 + 4x4 can't be added as the exponents are different.
  • 3x2 - 2y2 can't be subtracted as the bases are different.
  • 233 + 322 can be added only by evaluating 233 and 322 separately and then adding. So it is 18 + 12, which is 30.

Multiplying Exponents

  • To multiply exponents with the same base, keep the base and add the exponent.
  • ax × ay is the same as  ax +y.
  • 32 × 34 is the same as 32 + 4 which equates to 36.
  • If the bases are different, simplify each expression and then perform the multiplication.

Dividing Exponents

  • To divide exponents with the same base, keep the base and subtract the exponent.
  • ax ÷ ay is the same as ax - y.
  • 34 ÷ 32 is the same as 34 - 2 which equates to 32.
  • If the bases are different, simplify each expression and then perform the division.

Power of Exponents

  • To raise an exponent to a power, multiply the exponents.
  • 234 is the same as 212.
  • x3-2 is the same as  x-6.

Exponent Cheat Sheet

  • x0 = 1
  • x1 = 1
  • x-1 = 1x
  • am × an = am + n
  • aman = am - n
  • amn = am × n

Blunder Area

  • x2 + x3 can't be added directly as the exponents are different.
  • 23 + 33 can't be added directly as the bases are different.
  • x32 is x6 and not x5