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Chapter R. Section 2 Page 1 of 2 Section R.2 – Integer Exponents, Scientific Notation, Order of Operations Exponents: • For any positive integer n, a n = a ⋅ a ⋅ a " a (n times) . a is called the base and n is the exponent • When you have a negative exponent, swap the term across the fraction bar (if in the numerator move it to the denominator, and if in the denominator move to the numerator) 1 1 i.e. a − m = m , − n = a n a a • Anything (except 0) to the power of 0 is 1 (a0 = 1) • Properties of exponents: a m ⋅ a n = a m+ n (a m ) n = a mn (ab) m = a mb m • Using the above properties, what would be the following? • Example. Simplify a 0 ⋅ a 4 a 0 ⋅ a 4 = a 0+ 4 = a 4 Example. Simplify (−6b −4 )(2b −7 ) • am ⎛ a ⎞ , ⎜ ⎟ an ⎝ b ⎠ m −12b −4−7 = −12b −11 • ⎛ 125 p12 q −14 r 22 ⎞ Example. Simplify ⎜ 8 6 −15 ⎟ ⎝ 25 p q r ⎠ −4 −4 4 ⎛ 5 p 4 r 37 ⎞ ⎛ q 20 ⎞ q80 = = ⎜ 20 ⎟ ⎜ 4 37 ⎟ 16 148 ⎝ q ⎠ ⎝ 5 p r ⎠ 625 p r Scientific Notation: • Scientific notation for a number is an expression of the form N ×10m , where 1 ≤ N < 10 , N is in decimal form, and m is an integer • Large numbers have a positive exponent, and small numbers have a negative exponent. It has nothing to do with the sign of the number, only its size i.e. 4.5 ×104 = 45, 000 4.5 × 10−4 = 0.00045 • To convert to scientific notation, move the decimal place to the left (or right) and keep track of this value, until you reach a value of N between 1 and 10, this will be +m (moving left) or –m (moving right) • Example. Convert 1,670,000 to scientific notation 1670000 = 1.67 ×106 • Example. Convert 0.00000000514 to scientific notation 5.14 ×10−9 • To convert from scientific notation, write down N, move the decimal place m units (to the left if negative, to the right if positive) C. Bellomo, revised 26-Aug-07 Chapter R. Section 2 Page 2 of 2 • • • • Example. Convert 3.15 ×10−6 to decimal notation 0.00000315 Example. Convert 8.409 ×1011 to decimal notation 840,900,000,000 When computing with scientific notation, all the rules for exponents apply 1.3 × 104 Example. Compute 5.2 × 1010 1.3 × 10−6 = 0.25 × 10−6 = 2.5 × 10−7 5.2 Order of Operations: • It is essential to do computations in the proper order, or the result will be incorrect • Groupings, exponents, multiplication/division, addition/subtraction • Example, page 14 number 88. Calculate 3[(2 + 4 ⋅ 22 ) − 6(3 − 1)] 3[(2 + 4 ⋅ 4) − 6(3 − 1)] = 3[(2 + 16) − 6(2)] = 3[18 − 12] = 3[6] = 18 • Be sure you are familiar with how your calculator works as well, as many operate differently C. Bellomo, revised 26-Aug-07