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8.2 Negative and Zero Exponents I love exponents! Definition of Negative Exponents (let a be a nonzero number and let n be a positive integer) The expression a-n is the reciprocal of an . n 1 = a a-n = 1 a≠0 a-n an 3-2 = 1 32 =1 9 The negative exponent says the number needs to be moved to the opposite location and made positive. If it’s negative in the numerator, it belongs in the denominator position positive. If it’s negative in the denominator position, it belongs in the numerator position positive. Definition of Zero Exponent (let a be a nonzero number and let n be a positive integer) A nonzero number to the zero power is ALWAYS 1! a0 = 1 a 0 30 = 1 (x2y5)0 = 1 The expression 00 is undefined. Simplify expressions: write with positive exponents. (-5)-3 = 1 = -1 (-5)3 125 244-3 = 24 ∙ 1 = 24∙ 1 = 3 43 -3-4 = 1 = - 1 4 -3 81 1 64 8 Simplify expressions: write with positive exponents. 3a-3b-2 = 33 2 ab (3-3)2 = 3(-3 ∙ 2) = 3-6 = 1 1 = 6 3 729 ANOTHER WAY TO LOOK AT IT: (-3) squared = (-3)(-3) = 9 correct -3 squared = (-1)(3)(3) = -9 Graphing with a variable as an exponent Sketch the graph of y = 2x x 3 2 1 0 -1 -2 -3 2x 8 4 2 20=1 2-1= ½ 2-2 = ¼ 2-3 = 1/8 Will it ever touch the x-axis? Example: (Just follow this example to see what you get to do on your assignment.) Between 1970 and 1990, the Missouri population increased at a rate of .47% per year. The population P in t years is given by: P = 4,903,000 1.0047t Where t = 0 for 1980 Find the population in 1970, 1980, and 1990 To find the population, plug the numbers into the formula and then use a calculator. Pop in 1970 t = -10 Pop in 1990 t = 10 Do you expect it to be more or less in 1970 than 1980? P = 4,903,000 1.0047-10 (set up the problem) = 4,678,406 (calculate) This is the population in 1970. Population in 1990: P = 4,903,000 1.004710 P = 5,138,376