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Section 0.2 Integer Exponents and Scientific Notation Natural-Number Exponents When two or more quantities are multiplied together, each quantity is called a factor of the product. For any natural mumber n, xn = x · x · x · x · · · x • In the exponential expression xn , x is the base and n is the exponent or power to which the base is raised. The expression xn is called a power of x. NOTE: x1 = x. • The number 1 is not usually written. • Ex: Write each expression without using exponents: a. 32 = 9 b. (−3)2 = (−3)(−3) = 9 c. 2x4 = 2 · x · x · x · x d. (2x)4 = (2x)(2x)(2x)(2x) = 16 · x · x · x · x • NOTE: axn 6= (ax)n . axn = a · x · x · x · · · x while (ax)n = (ax)(ax)(ax) · · · (ax). Also, −xn 6= (−x)n . −xn = −x · x · x · · · x while (−x)n = (−x)(−x)(−x) · · · (−x). Rules of Exponents • Product Rule for Exponents If m and n are natural numbers, xm xn = xm+n . • NOTE: The product rule only applies to exponential expressions with the same base. • Power Rule of Exponents If m and n are natural numbers, then !n xn x m n mn n n n = n , (y 6= 0) (x ) = x (xy) = x y y y Ex: Simplify a. x3 x7 = x3+7 = x10 b. x3 y 3! x4 y 2 = x3+4 y 3+2 = x7 y 5 2 x2 x4 c. = y3 y6 • Zero Exponent x0 = 1, (x 6= 0). • Negative Exponents If n is an integer and x 6= 0, then x−n = 1 xn 1 = xn x−n • Quotient Rule for Exponents If m and n are integers, then xm m−n x (x 6= 0) = • A Fraction to a Negative Power If n is a natural number, then x y !−n = n y x (x 6= 0, y 6= 0) Ex: Simplify and write all answers without using negative exponents. x−6 1 a. 2 = x(−6)−2 = x−8 = 8 x x b. c. x5 x−3 = x5+(−3)−2 = x0 = 1 x2 3x2 y −5 2x−2 y −6 !−3 = 3x2 x2 y 6 2y 5 !−3 = 3x4 y 6 2y 5 !−3 = 3x4 y 2 !−3 = 2 3x4 y !3 = 8 27x12 y 3 Order of Operations Please Excuse My Dear Aunt Sally: (Parenthesis, Exponents, Multiplication/Division, Addition/ Subtraction) 1. Work from innermost to outermost grouping symbols (parenthesis or brackets). 2. Find the values of any exponential expressions. 3. Perform all multiplication and/or division from left to right. 4. Perform all addition and/or subtraction from left to right. 5. In a fraction, simplify the numerator and denominator separately. Then simplify. Evaluating Expressions Ex: If x = −2, y = 0, andz = 3, evaluate the expression: 5x2 − 3y 3 z = 5(−2)2 − 3(0)3 (3) = 5(4) − 3(0)(3) = 20 − 0 = 20 Scientific Notation • Scientific Notation A number is written in scientific notation when it is written in the form N x10n where 1 ≤ |N | < 10 and n is an integer. • We can write 372000 in scientific notation. To do so, we need to write it as the product of a number between 1 and 10 and some power of 10. The number 3.72 is between 1 and 10. We can change 3.72 into 372000 by moving the decimal point 5 places to the right. So, 3.72x105 = 372000. Using Scientific Notation to Simplify Computations Ex: Calculate (0.00000035)(170, 000) using scientific notation. 0.00000085 (3.5x107)(1.70x105) 5.95 (3.5)(1.7)(107+5 ) = = 7 7 8.5x10 (8.5)(10 ) 8.5 1012 107 ! = 0.7x105 = 7x104