Angel, Intermediate Algebra, 7ed 1 Aim: How do we simplify exponential expressions? Do Now: Simplify 1) 3⁴ 2) 2 · 3³ 3) 10 · 3² HW # 10 Chapter 7 pg 289 #25-26, pg 293 #57-63

Exponent Rules and Simplifying Exponential Expressions

Angel, Intermediate Algebra, 7ed 3 Rules of Exponents For all real numbers a and b and all rational numbers m and n, Product rule: a m a n = a m + n Quotient rule: Negative exponent rule:

Angel, Intermediate Algebra, 7ed 4 Rules of Exponents For all real numbers a and b and all rational numbers m and n, Zero exponent rule: a 0 = 1, a  0 Raising a power to a power: Raising a product to a power : Raising a quotient to a power :

Angel, Intermediate Algebra, 7ed 5  Simplify 1) 9⁴ · 3³ as a power of base 3 only 2) (4xy²)³ 3) (2x²y⁴)³ + (3x³y⁶)² 4) (5a³ b²)²(2a⁴b³)³

Angel, Intermediate Algebra, 7ed 6 Negative Exponents Use the exponent rules to try and solve: 1)s¯³ + s⁰ if s = 2 2)(s⁵t³)¯⁴ Express with only positive exponents 3)(a¯²b³)¯³ Express with only positive exponents 4)(r⅗)¯⁵ · (r¯⅔)³ positive exponents 5)(2x¯³y⅓)¯² · (8x⅔z¯²)³ positive exponents

Angel, Intermediate Algebra, 7ed 7 Rules of Exponents Examples : 1.) Simplify x -1/2 x -2/5. 2.) Simplify (y -4/5 ) 1/3. 3.) Multiply –3a -4/9 (2a 1/9 – a 2 ).

Angel, Intermediate Algebra, 7ed 8 Aim: How do we change radicals to rational exponents? Do Now: 1)Evaluate a² + 3a¯² - a⁰ if a = 2 2) Simplify (2a³b⁴c)⁵ 3)Simplify 36p⅓q⅘ (3p⅔q)² HW #11 pg 297 #40-45 pg 298 #74-77

Angel, Intermediate Algebra, 7ed 9 Changing a Radical Expression When a is nonnegative, n can be any index. When a is negative, n must be odd. A radical expression can be written using exponents by using the following procedure:

Angel, Intermediate Algebra, 7ed 10 Changing a Radical Expression When a is nonnegative, n can be any index. When a is negative, n must be odd. Exponential expressions can be converted to radical expressions by reversing the procedure.

Angel, Intermediate Algebra, 7ed 11 Simplifying Radical Expressions This rule can be expanded so that radicals of the form can be written as exponential expressions. For any nonnegative number a, and integers m and n, Power Index

Angel, Intermediate Algebra, 7ed 12  Try these: Rewrite using fractional exponents and simplify: 1) √4⁵ 2) ³√8⁴ 3) ³√27²  Rewrite in radical form and simplify 4) 64⅔ 5)32⅗ 6)81 - 3/4

Angel, Intermediate Algebra, 7ed 13 Try these: 1)Evaluate (x⅓ - x⁰ + 2x 3/2 ) if x =64 2) Evaluate 125 ¯ 4/3 3)Multiply (4a¯²b⅗)(2a¯³b⅚) 4) Rewrite with fractional exponents: a)√r³s b) ⁴√81a²b c) 7³√d⁵

Do Now: Write using a radical 1) 3a⅓ 2) (3a)⅓ Write in exponential form: 3) 7√5 4) 2(⁵√b) Evaluate: 5) 125 ¯⅓ 6) (-125)⅓ HW #12 Chapter 3 pg 93-94, #9,10,15,16,27,28 Aim: How do we simplify radicals?

Do Now: 1)Evaluate z⅕ + 2z⁰ + (2z)¯⅔ when z =32 2) Rewrite and simplify using positive exponents a) 12r¯³s⁵ b) 6p¯²q⁴ 20r⁶s 2¯³p⁵q¯² HW #12 Chapter 3 pg 93-94, #9,10,15,16,27,28 Aim: How do we simplify radicals?

Simplifying Radicals

Angel, Intermediate Algebra, 7ed 17 Definitions A perfect square is the square of a natural number. 1, 4, 9, 16, 25, and 36 are the first six perfect squares. Variables with exponents may also be perfect squares. Examples include x 2, (x 2 ) 2 and (x 3 ) 2. A perfect cube is the cube of a natural number. 1, 8, 27, 64, 125, and 216 are the first six perfect cubes. Variables with exponents may also be perfect cubes. Examples include x 3, (x 2 ) 3 and (x 3 ) 3.

Angel, Intermediate Algebra, 7ed 18 Perfect Powers A quick way to determine if a radicand x n is a perfect power for an index is to determine if the exponent n is divisible by the index of the radical. Example: Since the exponent, 20, is divisible by the index, 5, x 20 is a perfect fifth power. This idea can be expanded to perfect powers of a variable for any radicand. The radicand x n is a perfect power when n is a multiple of the index of the radicand.

Angel, Intermediate Algebra, 7ed 19 Product Rule for Radicals Examples:

Angel, Intermediate Algebra, 7ed 20 Try these, simplify: 1) √50 2)√16 3)√98c⁴

Angel, Intermediate Algebra, 7ed 21 Product Rule for Radicals 1.If the radicand contains a coefficient other than 1, write it as a product of the two numbers, one of which is the largest perfect power for the index. 2.Write each variable factor as a product of two factors, one of which is the largest perfect power of the variable for the index. 3.Use the product rule to write the radical expression as a product of radicals. Place all the perfect powers under the same radical. 4.Simplify the radical containing the perfect powers. To Simplify Radicals Using the Product Rule

Angel, Intermediate Algebra, 7ed 22 Product Rule for Radicals Examples: *When the radical is simplified, the radicand does not have a variable with an exponent greater than or equal to the index.

Angel, Intermediate Algebra, 7ed 23 Quotient Rule for Radicals Examples: Simplify radicand, if possible.

Angel, Intermediate Algebra, 7ed 24 Quotient Rule for Radicals More Examples:

Do Now: Simplify 1) 4√200 2) 5³√27a⁶b³ 3)√27x⁵y²³ HW #13 pg 97 #7,8,21 ;pg 101 #8,14,30,35 Aim: How do we add, subtract and multiply radicals?

Adding, Subtracting, and Multiplying Radicals

Angel, Intermediate Algebra, 7ed 27 Like Radicals Like radicals are radicals having the same radicands. They are added the same way like terms are added. Example: Cannot be simplified further.

Angel, Intermediate Algebra, 7ed 28 Adding & Subtracting Examples: 1.Simplify each radical expression. 2.Combine like radicals (if there are any). To Add or Subtract Radicals

Angel, Intermediate Algebra, 7ed 29 CAUTION! The product rule does not apply to addition or subtraction!

Angel, Intermediate Algebra, 7ed 30 Try These: 1)√27 + √12 2)³√16 - 5³√54 3)x√3 + √75x² 4)√⁴⁄₉ a²b³ + 3√a²b³

Angel, Intermediate Algebra, 7ed 31 Multiplying Radicals Multiply: Use the FOIL method. Notice that the inner and outer terms cancel.

Angel, Intermediate Algebra, 7ed 32 Try These: 1)6√5 ● 2√15 2)√3 (3√2 + √3) 3)(6 - √2) (4 + √2)

Do Now: 1) 6√5 ● 2√15 2) √3 (3√2 + √3) 3) (6 - √2) (4 + √2) 4) 3√45 + 5√80 5) √20/9 √5 Aim : What are conjugates and how do we rationalize denominators?

Angel, Intermediate Algebra, 7ed 34 Rationalizing Denominators Examples: To Rationalize a Denominator Multiply both the numerator and the denominator of the fraction by a radical that will result in the radicand in the denominator becoming a perfect power. Cannot be simplified further.

Angel, Intermediate Algebra, 7ed 35 Try These: Rationalize the denominator 1)1 √5 2)ab √b⁵

Angel, Intermediate Algebra, 7ed 36 Conjugates When the denominator of a rational expression is a binomial that contains a radical, the denominator is rationalized by multiplying both numerator and denominator by the conjugate. The conjugate of a binomial is a binomial having the same two terms with the sign of the second term changed. The conjugate of *Use FOIL when multiplying by the conjugate

Angel, Intermediate Algebra, 7ed 37 Simplifying Radicals Simplify by rationalizing the denominator: = 5(√2 – 1)

Angel, Intermediate Algebra, 7ed 38 Try These: Find the conjugate for : 1)3 + √2 2)√5 – 23 3) 4 – 2√7 Rationalize the denominator using the conjugate: 1) 3 2) 12 3) √2 √ √5