UNIT 2 – QUADRATIC, POLYNOMIAL, AND RADICAL EQUATIONS AND INEQUALITIES Chapter 6 – Polynomial Functions 6.1 – Properties of Exponents

6.1 – PROPERTIES OF EXPONENTS  In this section we will review:  Using properties of exponents to multiply and divide monomials  Using expressions written in scientific notation

6.1 – PROPERTIES OF EXPONENTS  To simplify an expression containing powers means to rewrite the expression without parentheses or negative exponents  Negative exponents are a way of expressing the multiplicative inverse of a number  1/x 2 = x -2

6.1 – PROPERTIES OF EXPONENTS  Negative Exponents  For any real number a ≠ 0 and any integer n, a –n = 1 / a n  2 -3 = 1 / 2 3 = 1 / 8  1 / b -8 = b 8

6.1 – PROPERTIES OF EXPONENTS  Example 1  Simplify each expression  (-2 a 3 b)(-5 ab 4 )  (3a 5 )(c -2 )(-2a -4 b 3 )

6.1 – PROPERTIES OF EXPONENTS  Product of Powers  For any real number a and integers m and n, a m · a n = a m + n  4 2 · 4 9 = 4 11  b 3 · b 5 = b 8  To multiply powers of the same variable, add the exponents.

6.1 – PROPERTIES OF EXPONENTS  Quotient of Powers  For any real number a ≠ 0, and any integers m and n, a m / a n = a m – n  5 3 / 5 = 5 3 – 1 = 5 2 and x 7 /x 3 = x 7 – 3 = x 4  To divide powers of the same base, you subtract exponents

6.1 – PROPERTIES OF EXPONENTS  Example 2  Simplify s 2 / s 10. Assume that s ≠ 0.

6.1 – PROPERTIES OF EXPONENTS  Properties of Powers  Suppose a and b are real numbers and m and n are integers. Then the following properties hold.  Power of a Power: (a m ) n = a mn  (a 2 ) 3 = a 6  Power of a Product: (ab) m = a m a m  (xy) 2 = x 2 y 2  Power of a Quotient: (a / b) n = a n / a n, b ≠ 0  (a / b) 3 = a 3 / b 3  Power of a Quotient: (a / b) -n = (b / a) n or b n / a n, a ≠0, b ≠0  (x / y) -4 = y 4 / x 4  Zero Power: a 0 = 1, a ≠ 0

6.1 – PROPERTIES OF EXPONENTS  Example 3  Simplify each expression  (-3c 2 d 5 ) 3  (-2a / b 2 ) 5

6.1 – PROPERTIES OF EXPONENTS  Example 4  Simplify (-3a 5y / a 6y b 4 ) 5

6.1 – PROPERTIES OF EXPONENTS  Standard notation – form in which numbers are usually written  Scientific Notation – a number in form a x 10 n, where 1 ≤ a < 10 and n is an integer.  Real world problems using numbers in scientific notation often involve units of measure.  Performing operations with units is know as dimensional analysis

6.1 – PROPERTIES OF EXPONENTS  Example 5  There are about 5 x 10 6 red blood cells in one milliliter of blood. A certain blood sample contains 8.32 x 10 6 red blood cells. About how many milliliters of blood are in the sample?

6.1 – PROPERTIES OF EXPONENTS HOMEWORK Page 316 #11 – 37 odd