Just-In-TimeReview Sections 22-25
JIT22: SimplifyRadicalEx- pressions
Definition of n th Roots √ x = y . If n is a natural number and y n = x , then n
Definition of n th Roots √ x = y . If n is a natural number and y n = x , then n √ x is read as “the n th root of x ”. n
Definition of n th Roots √ x = y . If n is a natural number and y n = x , then n √ x is read as “the n th root of x ”. n √ 8 = 2 because 2 3 = 8. 3
Definition of n th Roots √ x = y . If n is a natural number and y n = x , then n √ x is read as “the n th root of x ”. n √ 8 = 2 because 2 3 = 8. 3 √ 81 = 3 because 3 4 = 81. 4
Definition of n th Roots √ x = y . If n is a natural number and y n = x , then n √ x is read as “the n th root of x ”. n √ 8 = 2 because 2 3 = 8. 3 √ 81 = 3 because 3 4 = 81. 4 When n = 2 we call it a “square root”. However, instead of writing √ x , we drop the 2 and just write √ x . So, for example: 2
Definition of n th Roots √ x = y . If n is a natural number and y n = x , then n √ x is read as “the n th root of x ”. n √ 8 = 2 because 2 3 = 8. 3 √ 81 = 3 because 3 4 = 81. 4 When n = 2 we call it a “square root”. However, instead of writing √ x , we drop the 2 and just write √ x . So, for example: 2 √ 16 = 4 because 4 2 = 16.
Domain of Radical Expressions How do you find the even root of a negative number? For example, √− 16 is the number x . 4 imagine the answer to
Domain of Radical Expressions How do you find the even root of a negative number? For example, √− 16 is the number x . 4 imagine the answer to This means x 4 = − 16. The problem is, if we raise a number to the fourth power, we never get a negative answer: (2) 4 = 2 · 2 · 2 · 2 = 16 ( − 2) 4 = ( − 2)( − 2)( − 2)( − 2) = 16
Domain of Radical Expressions How do you find the even root of a negative number? For example, √− 16 is the number x . 4 imagine the answer to This means x 4 = − 16. The problem is, if we raise a number to the fourth power, we never get a negative answer: (2) 4 = 2 · 2 · 2 · 2 = 16 ( − 2) 4 = ( − 2)( − 2)( − 2)( − 2) = 16 Since there is no solution here for x , we say that the fourth root is undefined for negative numbers.
Domain of Radical Expressions How do you find the even root of a negative number? For example, √− 16 is the number x . 4 imagine the answer to This means x 4 = − 16. The problem is, if we raise a number to the fourth power, we never get a negative answer: (2) 4 = 2 · 2 · 2 · 2 = 16 ( − 2) 4 = ( − 2)( − 2)( − 2)( − 2) = 16 Since there is no solution here for x , we say that the fourth root is undefined for negative numbers. However, the same idea applies to any even root (square roots, fourth roots, sixth roots, etc) √ n If the expression contains B where n is even, then B ≥ 0 .
Properties of Roots/Radicals √ x n = x if n is odd. n
Properties of Roots/Radicals √ x n = x if n is odd. n √ x n = | x | if n is even. n
Properties of Roots/Radicals √ x n = x if n is odd. n √ x n = | x | if n is even. n √ x n √ xy = √ y n n (When n is even, x and y need to be nonnegative.)
Properties of Roots/Radicals √ x n = x if n is odd. n √ x n = | x | if n is even. n √ x n √ xy = √ y n n (When n is even, x and y need to be nonnegative.) √ x � n x y = n √ y n (When n is even, x and y need to be nonnegative.)
Properties of Roots/Radicals √ x n = x if n is odd. n √ x n = | x | if n is even. n √ x n √ xy = √ y n n (When n is even, x and y need to be nonnegative.) √ x � n x y = n √ y n (When n is even, x and y need to be nonnegative.) √ √ x x m = � m � n n (When n is even, x needs to be nonnegative.)
Properties of Roots/Radicals √ x n = x if n is odd. n √ x n = | x | if n is even. n √ x n √ xy = √ y n n (When n is even, x and y need to be nonnegative.) √ x � n x y = n √ y n (When n is even, x and y need to be nonnegative.) √ √ x x m = � m � n n (When n is even, x needs to be nonnegative.) √ x = √ x � m n mn
Examples √ √− 81 3 3 1. 375 +
Examples √ √− 81 3 3 1. 375 + √ 3 2 3
Examples √ √− 81 3 3 1. 375 + √ 3 2 3 √ 2. 16 x 2
Examples √ √− 81 3 3 1. 375 + √ 3 2 3 √ 2. 16 x 2 4 | x |
Examples √ √− 81 3 3 1. 375 + √ 3 2 3 √ 2. 16 x 2 4 | x | � � 3. 5 x 3 y 4 5 x 7 y
Examples √ √− 81 3 3 1. 375 + √ 3 2 3 √ 2. 16 x 2 4 | x | � � 3. 5 x 3 y 4 5 x 7 y x 2 y
Examples √ √− 81 3 3 1. 375 + √ 3 2 3 √ 2. 16 x 2 4 | x | � � 3. 5 x 3 y 4 5 x 7 y x 2 y � 16 ab 4. 3 a 4 b 3
Examples √ √− 81 3 3 1. 375 + √ 3 2 3 √ 2. 16 x 2 4 | x | � � 3. 5 x 3 y 4 5 x 7 y x 2 y � 16 ab 4. 3 a 4 b 3 � 2 2 3 a b 2
JIT23: RationalizingNu- meratorsandDenominators
Definition Rationalizing the denominator of a fraction is simplifying the fraction so that the denominator doesn’t have any roots or radicals.
Definition Rationalizing the denominator of a fraction is simplifying the fraction so that the denominator doesn’t have any roots or radicals. Rationalizing the numerator of a fraction is simplifying the fraction so that the numerator doesn’t have any roots or radicals.
Methods The numerator/denominator has The numerator/denominator has no addition or subtraction: addition or subtraction: Simplify any roots as much To rationalize, multiply the as possible. top and bottom of the fraction by the conjugate - The numerator/ the expression you get when denominator you’re trying you flip the sign in the to rationalize should have √ “middle”. n x m . a √ x + b √ y ← → a √ x − b √ y Multiply the top and bottom of the fraction by √ n x n − m .
Examples 1. Rationalize the denominator: 3 √ 2 − 7
Examples 1. Rationalize the denominator: 3 √ 2 − 7 √ − 2 − 7
Examples 1. Rationalize the denominator: 3 √ 2 − 7 √ − 2 − 7 2. Rationalize the numerator: √ √ 3 + 2 √ √ 3 − 4 2
Examples 1. Rationalize the denominator: 3 √ 2 − 7 √ − 2 − 7 2. Rationalize the numerator: √ √ 3 + 2 √ √ 3 − 4 2 1 √ 11 − 5 6
Examples 3. Rationalize the 1. Rationalize the denominator: denominator: � 3 3 3 √ 5 2 − 7 √ − 2 − 7 2. Rationalize the numerator: √ √ 3 + 2 √ √ 3 − 4 2 1 √ 11 − 5 6
Examples 3. Rationalize the 1. Rationalize the denominator: denominator: � 3 3 3 √ 5 2 − 7 √ √ 3 75 − 2 − 7 5 2. Rationalize the numerator: √ √ 3 + 2 √ √ 3 − 4 2 1 √ 11 − 5 6
Examples 3. Rationalize the 1. Rationalize the denominator: denominator: � 3 3 3 √ 5 2 − 7 √ √ 3 75 − 2 − 7 5 2. Rationalize the numerator: 4. Rationalize the numerator: √ √ 3 + 2 √ √ √ 7 3 − 4 2 2 1 √ 11 − 5 6
Examples 3. Rationalize the 1. Rationalize the denominator: denominator: � 3 3 3 √ 5 2 − 7 √ √ 3 75 − 2 − 7 5 2. Rationalize the numerator: 4. Rationalize the numerator: √ √ 3 + 2 √ √ √ 7 3 − 4 2 2 1 7 √ √ 11 − 5 6 2 7
JIT24: RationalExponents
Definition of Rational Exponents Every n th root has an equivalent exponential form: √ x = x 1 / n n When roots are written in their exponential form, you can use all of the properties of exponents to simplify problems. Another nice formula to convert is √ x m = x m / n n
Examples Simplify the following expressions, writing your final answer without negative exponents. Assume all variables denote positive quantities. √ √ 3 a 4 b − 1 9 1. a 6 b 2
Examples Simplify the following expressions, writing your final answer without negative exponents. Assume all variables denote positive quantities. √ √ 3 a 4 b − 1 9 1. a 6 b 2 a 2 b 1 / 9
Examples Simplify the following expressions, writing your final answer without negative exponents. Assume all variables denote positive quantities. √ √ 3 a 4 b − 1 9 1. a 6 b 2 a 2 b 1 / 9 � − 1 / 4 � r 8 s − 4 2. 16 s 4 / 3
Examples Simplify the following expressions, writing your final answer without negative exponents. Assume all variables denote positive quantities. √ √ 3 a 4 b − 1 9 1. a 6 b 2 a 2 b 1 / 9 � − 1 / 4 � r 8 s − 4 2. 16 s 4 / 3 4 / 2 s 3 r 2
JIT25: PythagoreanTheo- rem
Formula The Pythagorean Theorem is a formula that relates the lengths of the sides of a right triangle (a triangle where one of the angles is 90 ◦ ). a 2 + b 2 = c 2 c b a
Examples Find the length of the side not given. c b a 1. a = 4, b = 6
Examples Find the length of the side not given. c b a 1. a = 4, b = 6 √ 2 13
Examples Find the length of the side not given. c b a 1. a = 4, b = 6 2. a = 3, c = 5 √ 2 13
Examples Find the length of the side not given. c b a 1. a = 4, b = 6 2. a = 3, c = 5 √ 4 2 13
Recommend
More recommend
