4.1 Polynomials

4.1 Natural-Number Exponents

Objectives Learn the meaning of exponential notation Simplify calculation by using product rule for exponents Simplify calculation by using power rule for exponents Simplify calculation by using quotient rule for exponents

Japanese Sword Making Japanese sword making Begins with raw iron (1:15-3:07) Forged and folded 15 times. (4:34-8:06) How many layers does this produce? 2 15 = 32,768

Meaning of Exponential Notation Note: a 3 = a  a  a a.2 5 = 2  2  2  2  2 = 32 b.(–7) 3 = (–7)(–7)(–7) = –343 c.–y 5 = –y  y  y  y  y d.-5 3 = -(5  5  5) = -125

Meaning of Exponential Notation If n is a natural number, then x n = x  x  x  …  x x n n factors of x Base Exponent

Examples a. 3 1 = 3 b. (–y) 1 = –y c. (–4z) 2 = (–4z)(–4z) = 16z 2 d. (t 2 ) 3 = t 2  t 2  t 2 = t 6

Examples Show that (–2) 4 and –2 4 have different values. o (–2) 4 = (-2)(-2)(-2)(-2) = 16 o –2 4 = -(2  2  2  2) = -16

Examples Simplify (-2) 3 3.(-3) General Rule ◦ If the exponent is even, result is positive. ◦ If the exponent is odd, result is the same sign as that of the original base.

Product Rule for Exponents We note: x 2 x 3 = x  x  x  x  x = x  x  x  x  x = x 5 Product Rule for Exponents x m  x n = x  x  x ...  x  x  x  x ...  x = x  x  x  x  x  x ...  x  x  x = x m + n m factors of x n factors of x m + n factors of x

Product Rule for Exponents If m and n are natural numbers {1, 2, 3, …}, then x m x n = x m + n

Power Rule for Exponents If m and n are natural numbers {1, 2, 3, …}, then (x m ) n = x mn Example a. (2 3 ) 7 = 2 3  7 = 2 21 b. (z 7 ) 7 = z 7  7 = z 49

Example Simplify (2x) 3 (2x) 3 = (2x)(2x)(2x) = (2  2  2)(x  x  x) = 2 3 x 3 = 8x 3

Examples Simplify each expression. a.x 3 x 4 = x = x 7 b.y 2 y 4 y = ( y 2 y 4 )y = ( y )y = y 6 y = y = y 7

Power Rules for Exponents We note: (x 3 ) 4 = x 3  x 3  x 3  x 3 = x  x  x  x  x  x  x  x  x  x  x  x = x 12 This suggests the Power Rule for Exponents. (x m ) n = x m  x m  x m ...  x m = x  x  x  x  x  x  x ...  x = x m  n 12 factors of x X3X3 x3x3 x3x3 x3x3 m  n factors of x

Your Turn Simplify: a.(2 3 ) 7 = 2 3  7 o = 2 21 b. (z 7 ) 7 = z 7  7 o = z 49 C.(2x) 3 = (2x)(2x)(2x) o = (2  2  2)(x  x  x) = 2 3 x 3 = 8x 3

Example Simplify.

Quotient Rule for Exponents Note: This suggests: = = 4 3

Quotient Rule for Exponents Quotient Rule for Exponents If m and n are natural numbers, m  n and x  0, then

Examle Simplify. Assume no division by zero.

Example