Chapter 2 : Powers and Exponents



Power arrow_upward


  • Power is the number of times a number or expression is multiplied by itself.
  • Another name for power is index or exponent.
  • Examples :
    • 25 = 2 × 2 × 2 × 2 × 2 = 32
    • (-4)3 = (-4) × (-4) × (-4) = -64
    • 1n = 1 for any n, 0n = 0 for any n

    Squares and Cubes arrow_upward


  • bn is read as “b raised to the nth power”.
  • b2 is read as “b squared”.
  • b2 is always greater than 0 (positive) if b is not zero, since the product of two negative numbers is positive.
  • b3 is read as “b cubed”.
  • b3 can be negative or positive.
  • The following table shows squares and cubes of numbers from 1 to 10:

  • Number

    Square

    Cube

    1

    12 = 1

    13 = 1

    2

    22 = 4

    23 = 8

    3

    32 = 9

    33 = 27

    4

    42 = 16

    43 = 64

    5

    52 = 25

    53 = 125

    6

    62 = 36

    63 = 216

    7

    72 = 49

    73 = 343

    8

    82 = 64

    83 = 512

    9

    92 = 81

    93 = 729

    10

    102 = 100

    103 = 1000


  • If you raise a fraction, p/q, to a power, then  
  • Example:

  • Exponent arrow_upward


  • An exponent is used to show repeated multiplication.
  • Where,
    • a and nare integers.
    • The integera is called the base andn is called the exponent.
  • In the expression bn , b is called the base and n is called the exponent.
  • In the expression 25 , 2 is the base and 5 is the exponent.

  • Rules of Exponents arrow_upward


  • The exponent says how many times to multiply the given number by itself.
  • A negative exponent means divide.
  • A fractional exponent like 1/n means to take the nth root:

  • Rules

    Example

    x1 = x

    71 = 7

    x0 = 1

    80 = 1

    x-1 = 1/x

    4-1 = 1/4

    xm xn = xm+n

    x3 x5 = x3+5 = x8

    (xm )n = xmn

    (x4 )5 = x4×5 = x20

    (xy)n = xn yn

    (xy)7 = x7 y7



    Exponents: Properties arrow_upward


     

  • Example: What is the value of 63 ?
  • Solution: Since 6 = 3 × 2
  •     = 27 × 8

        = 216


    Negative Exponents arrow_upward


  • For any non-zero integer b,
  •  
  • so
  • Multiplying by b-n is same as dividing by bn .
  • Example:
  • Example: Find the value of ?
  • Solution:

  • Exponential Expansions arrow_upward



    Function

    Summation Expansion







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