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Chapter 3 Powers and Exponents Section 3.1 Using Exponents to Describe Numbers Exponents are used to write repeated multiplications by the same number in a more compact form. Eg.) 5x5x5x5 = 54 54 is in exponential form, where 5 is the base (the number being multiplied by itself) and 4 is the exponent – tells you the number of times you multiply the base by itself. 54 is loosely called a power (a base and a power). It is often read in several ways: 5 to the fourth 5 to the fourth power 5 to the exponent four Some common forms: 32 (three squared) 1) 3 3x3=9 3 (two cubed) 2) 2 2 2 2x2x2=8 3) 54 on a calculator: 5 yx 4 = 625 (standard form) 4) Some can be written 36 = (3x3) x(3x3) x (3x3) = 9 X 9 X 9 = 93 = 729 OR (3X3X3) X (3X3X3) = 27 X 27 = 272 = 729 Powers with negative bases: Eg #1) (‐2)4 – base is negative (‐2)(‐2)(‐2)(‐2) = +16 ‐ positive answer because there are an even amount of negatives. Eg #2) ‐24 ‐ without the bracket, only the 2 is the base. ‐ (2)(2)(2)(2) = (‐16) ↑ treat this as a (‐1) and multiply Eg #3) (‐2)3 = (‐2)(‐2)(‐2) = (‐8) – negative because there’s an odd number for an exponent. *Note: the exponent only applies to the base, which is the fist number at it’s left, or what is in the brackets. ‐(‐5)6 ‐ base is (‐5) , but not the negative in front. = ‐ (‐5)(‐5)(‐5)(‐5)(‐5)(‐5) = ‐ (+15625) = ‐15,625 Prime Factorization: 36 6 6 2 3 2 3 Section 3.2 Exponent Laws Factored form is another name for repeated multiplication. 1.) am x an = am+n **when multiplying same base, add exponents. Eg) 22 x 23 = 25 (2x2)(2x2x2)=25 m 2) am ÷ an or a a n = am‐n **when dividing same bases, subtract exponents. Eg) 24 ÷ 22 = 22 or 2x2x2x2 = 22 2x2 € 3) (am)n = amxn **when a power with an exponent is raised (taken) to another € power, multiply exponents. Eg) (22)3 = (22)(22)(22) = (2x2)(2x2)(2x2) = 26 4) (ab)m = ambm ** when a product (multiplied together) is raised to a power, the power applies to each factor separately. Eg) (‐2x3)3 = (‐2x3)(‐2x3)(‐2x3) = (‐2x‐2x‐2)(3x3x3) = (‐2)3(3)3 = ‐8(27) = ‐216 5) (a/b)n = an/bn **where a quotient (division) is raised to a power, rewrite with the exponent in each place.**b cannot equal 0. Eg) (2/3)3 = 23/33 = (2/3)(2/3)(2/3) = 23/33 = 8/27 6) a0 = 1 ** always equals 1, except a cannot equal 0 Eg) 103 = 1000 102 = 100 101 = 10 100 = 1 Section 3.3 Order of Operations Terminology Coefficient – is a number that multiplies an expression. (This can be another number or variable.) This needs to include the sign. Eg #1) ‐3(4)2 Coefficient = ‐3 Eg #2) ‐3x4 Coefficient = ‐3 Eg #3) 5x2y4 Coefficient = 5 Eg #4) (‐2)2 Coefficient = 1 (multiplicative identity) Eg #5) x2y Coefficient = 1 B Brackets E Exponents D Division M Multiplication A Addition S Subtraction **For a complex question (many operations) do only 1 step and rewrite the question, continue until solved. Negative example: ‐22 = ‐1(22) = ‐4 (‐2)2 = (‐2x‐2) = +4