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Unformatted text preview: The Power of Compounding Friday January 28, 2011 (continued) Review: Powers and Roots (a) Basics of Powers A number raised to the n th power is that number multiplied by itself n times ( n is called the exponent). For example 2 1 = 2 2 2 = 2 × 2 = 4 2 3 = 2 × 2 × 2 = 8 Any number (other than zero) to the zero power is one. For example, 2 = 1 . Negative powers are the reciprocals of the corresponding positive powers. For example 5 2 = 1 5 2 = 1 5 × 5 = 1 25 2 3 = 1 2 3 = 1 2 × 2 × 2 = 1 8 (b) Power Rules • To multiply powers of the same number, add the exponents. x n × x m = x n + m example: 2 3 × 2 2 = 2 5 = 32 • To divide powers of the same number subtract the exponents. x n x m = x n m example: 5 3 5 2 = 5 3 2 = 5 1 = 5 example: 5 2 5 3 = 5 2 3 = 5 1 = 1 5 1 = 1 5 • When a power is raised to another power, multiply the exponents: ( x n ) m = x n × m example: (2 2 ) 3 = 2 6 = 64 1 (c) Basics of Roots Finding a root is the reverse of raising a number to a power. Second roots (i.e. square roots) are written as a number under the symbol √ . Higher order roots are written as numbers under the symbol n √ . For example √ 4 = 2 because2 2 = 2 × 2 = 4 3 √ 27 = 3 because3 3 = 3 × 3 × 3 = 27 4 √ 16 = 2 because2 4 = 2 × 2 × 2 × 2 = 16 6 p 1 , 000 , 000 = 10 because10 6 = 1 , 000 , 000 (d) Roots as Fractional Powers The n th root of a number is the same as the number raised to the 1 /n power: n √ x = x 1 /n For example: 64 1 / 3 = 3 √ 64 = 4 1 , 000 , 000 1 / 6 = 6 p 1 , 000 , 000 = 10 Simple vs. Compound Interest Definitions In a financial formula, principal is the original amount of money invested. Simple Interest is interest paid only on the original principal. Compound Interest is interest paid on both the original principal and on all interest that has been added to the principal. Example 1. Simple Interest Suppose you deposit $ 1000 into an account that pays 5% simple interest per year. At the end of the first year, you get a payment of 5% × $1 , 000 = 0 . 05 × $1 , 000 = $50 . At the end of the second year, you get a payment of 5% × $1 , 000 = 0 . 05 × $1 , 000 = $50 . Similarly, at the end of each year, you receive a payment of $50. After 3 years you will have received a total of 3 × $50 = $150 in interest payments. This is an example of simple interest. At the end of each year, the interest payment you receive is %5 of the principal. Example 2. Compound Interest Suppose you deposit $1,000 into an account that pays %5 compound interest once per year. After the first year, you’ll receive a payment of 5% × $1 , 000 = 0 . 05 × $1 , 000 = $50 . With compound interest, the $50 you received as interest payment after year 1 will also have interest paid on it. So, at the end of year 2, you’ll receive an interest payment of %5 × $1 , 050 = 0 . 05 × $1 , 050 = $52 . 50 2 Adding this interest payment to your balance, increases your balance to $1 , 050 + %52 . 50 = $1102 . 50 ....
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 Spring '08
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