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Negative and rational exponents. =. =. 1. 1. 1. 1. b × b. b × b × b × b. b × b. x n. b 2. b 2. x – n =. Look at the following division:. b 2 ÷ b 4 =. Negative exponents. 1. Write an equivalent expression for the term using the division law for exponents. b 2.
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= = 1 1 1 1 b × b b × b × b × b b × b xn b2 b2 x–n= Look at the following division: b2 ÷ b4 = Negative exponents 1 Write an equivalent expression for the term using the division law for exponents. b2 Using the exponent law xm÷xn= x(m– n) b2 ÷ b4 = b(2 – 4) = b–2 This means that b–2 = In general,
Reciprocals 1 a b The reciprocal of a is a b a The reciprocal of is x-1 We can find reciprocals on a calculator using the key. A nonzero number raised to the power of –1 gives us the reciprocalof that number. The reciprocal of a number is what we multiply the number by to get 1.
Annual or monthly interest Riley puts $1000 in a savings account. The first $100 does not gather interest and the rest earns interest at 3.5% per year. Write a function for Riley’s total savings after t years. What is the equivalent monthly interest rate? Rewrite the function using this rate. f(t) = 100 + 900(1.035)t The annual interest rate, 1.035, is the monthly rate (m) multiplied by itself 12 times. This means that 1.035 = m12. f(t) = 100 + 900(m12)t m= 1.0351/12 f(t) = 100 + 900(m)12t The monthly rate is 0.29%. f(t) = 100 + 900(1.0351/12)12t ≈ 100 + 900(1.0029)12t
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 3 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 9 × 9 Suppose we have 9 . What is the value of ? Using the exponent law, 91 = 9 9×9= 9+= Rational exponents But also, 9 × 9 = 3 × 3 = 9 The square root of x. So in general, x = x Suppose we have 8 . What is the value of 8×8 ×8 ? 8 + + = 81 = 8 Using the exponent law, 8×8 ×8 = But also, 8 × 8 × 8 = 2 × 2 × 2 = 8 3 3 3 The cube root of x. So in general, x = x 3
1 n m n n x = x In general, Can you find a rule for expressions in the form x ? Rational exponents m 1 We can write x as x ×m n n and we know that (xm)n=xmn, 1 1 n x × m= (x)m = ( x )m therefore, n n m 1 Also, we can write x as xm× n n 1 1 n therefore, x = (xm) = xm n n m × m m or n n In general, x = xm x = (x )m n n
3 3 2 2 m m n n What is the value of 25 ? or n n In general, x = xm x = (x )m Rational exponents So using the 2nd rule we can write: 25 is a perfect square so it is easiest to use the rule where we square root first, then cube. 25 = (25) 3 = (5)3 = 125 Notice that the numerator of the fraction determines the powerand the denominator determines the degree of the root.