Review of Exponential Notation: Lesson 2 - read to the power of, where is the base and is the exponent - if no exponent is denoted, it is understood to be a power of 1 - if no coefficient is denoted, it is also understood to be 1 When the exponent is a positive integer, exponential notation is a concise way of writing the product of repeated factors. Product Rule for Exponents:, times if When common bases are multiplied, the exponents are added. ( a) m m m m ( b) 9 9 9 9 4 8 3 9 6 7 14 Quotient Rule for Exponents: When common bases are divided, the exponents are subtracted (exponent in numerator minus exponent in denominator) 3 3 9 3 3 ( a) r r ( b) x yz x y z x y z 8 4 4 12 2 6 8 2 6 8 4 2 5 4 8r s 8s 12x y z 4 4 Power Rule for Exponents: When a base is raised to a power and then raised to another power, the exponents are multiplied. 3 4 12 (4 ) 4 (2 x yx ) (2 x y) 8x y 3 2 4 3 3 6 3 18 3 4 2 4 4ab 2 4 4 8 3 2ab 16a b 2a 1
Product to a Power Rule: When a product is raised to a power, the exponent is applied to each factor. Keep in mind the difference between and. With, only is being raised to the power of, the coefficient is simply being multiplied by the result of. With, both the coefficient and the variable are being raised to the power of. ( 2 ab) 16 a b (3 xy ) 27x y 4 4 4 2 3 3 6 Quotient to a Power Rule: When a quotient is raised to a power, the exponent is applied to each factor in the numerator and denominator. Be sure to apply the exponent to EVERY factor inside the parentheses, both variables and coefficients. 4 4 256 xy x y 3 4 12 Zero-Exponent Rule: Any base (other than a base 0) taken to the power of zero is 1. (Zero to the zero power is not defined.) 2 2 (3 ) 0 2 2 0 a x y y b xy y y 2
Negative Exponent Rule: For a quantity to a negative power, to change the sign from negative to a positive exponent, take the reciprocal of the entire quantity to the positive exponent. Keep in mind that the sign of the base does NOT change. Only the sign of the exponent changes. 2 6 5 2 5 10 2 4 8 1 mn a m 2 n n ( a) ( a ) a ( b) ( c) 8 6 2 2 2 2 a a b b n n 2 32 For the next 4 examples, simplify the expression using the rules of exponents. 4 1 4 4 2 3 5 1 2a b a Ex 1) x y 4xy Ex 2) 8 4 3b ba 2 3
2 5 2 2 3 3 3 3x yz x z 2x Ex 3) Ex 4) 2 2 3 4 2 3 x y z y x y 1 4 2 Radical Notation: - read the root of, where is the radicand, is the index and is the radical sign - if no index is denoted, it is understood to be an index of 2 (square root) o - think of this as undoing radicals undo exponents, and exponents undo radicals (inverses) o Since o Then Definition of - the value that must be multiplied by itself times to produce ( ) ( ) ( ) Radicals and exponents are inverses (they undo each other) 4
When working with radicals that have an even index, only nonnegative numbers can go in and only nonnegative numbers can come out. You can only find an even root of a positive number or zero. (The positive root that comes out is called the PRINCIPAL root. The negative root should only be given as an answer if directions indicate by +/- or -.) When working with radicals that have an even index: - if a positive number goes in, an only a positive number comes out. - if a negative number goes in, there is no real root. - if a zero goes in, a zero comes out. - You can only find an even root of a positive number or zero. 4 6 16 2 64 2 121 11 8 0 0 36 is not real 16 4, 4 When working with radicals that have an odd index: - if a negative number goes in, a negative number comes out. - if a positive number goes in, a positive number comes out. - if zero goes in, zero comes out. - You can find an odd root of any real number. 27 3 32 2 1 1 8 2 3 5 11 3 Example 5: Evaluate each expression (if possible). a. = b. c. b. c. d. = e. f. 5
In the previous example, we found the following: Re-writing the radicands as a base to a power, we have the following: How can we re-write a radical using an exponent? (Hint: think of the Power Rule for Exponents) As can be noted above, the denominator of a rational (fractional) exponent indicated a root or radical. Example 6: Evaluate the following. 1 1 1/2 4 1/3 2 a) 9 b) 81 c) 125 d) ( 4) 1 1/2 3 e) 100 f) ( 8) Definition of : - the radical expressions and ( ) are equivalent to, where is a positive integer greater than and exists - it is imperative that exists, otherwise the expression is meaningless 6
Example 7: Express each number in the form,where and are integers. (Express as a fraction even if you write a 1 for the denominator.) a. b. c. d. Example 8: Simplify completely. Keep in mind, the Laws of Exponents are true for rational (fractional) exponents as long as the bases are positive or as long as the equivalent radical is defined. a. b. c. Product Rule for Radicals: - when indices are the same, radicands can be multiplied (if roots exist) - this can be used to multiply radicals or simplify radicals o Assume variables represent positive o numbers! - if the indices are not the same, you can try to simplify a radical expression using rational exponents (for example: ( )( ) ) 7
Quotient Rule for Radicals: - when indices are the same, radicands can be divided (if roots exist) - this can be used to divide radicals or simplify radicals o o Assume variables represent positive numbers! - if the indices are not the same, you can try to simplify the radical expression using rational exponents (for example: ) ***There is no Sum Rule for Radicals or Difference Rule for Radicals*** -, BUT - when a sum (difference) is raised to a power, the exponent is NOT applied to each term Simplifying radicals: - remove factors from the radical until no factor in the radicand has a degree greater than or equal to the index - use Product Rule for Radicals and/or Quotient Rule for Radicals Example 9: Simplify the expression completely. (Assume that all variables are positive.) a. b. b. 8
c. d. Example 10: Rewrite the expression using a radical, and simplify completely. a. b. AGAIN, There is no Sum or Difference Rule for Radicals. Rationalizing denominators: - re-writing a fraction so that the denominator contains only rational numbers or expressions (no radicals) 9
Steps for rationalizing denominators: 1. use quotient rule (if necessary) 2. simplify radical in the denominator 3. multiply by 1 ( ) I recommend you simplify any radical, if possible, before rationalizing a denominator. Example 11: Simplify the expression completely. (Assume that all variables are positive real numbers.) a. b. c. d. 10