1 / 25

Algebra II Chapter 7 7-1 Roots and Radical Expressions

Algebra II Chapter 7 7-1 Roots and Radical Expressions. MA.A.1.4.1, MA.A.1.4.4, MA.A.3.4.1. Vocabulary. N th root : for any real numbers a and b , and any positive integer n , if a n = b , then a is an n th root of b

natalie-brown
Télécharger la présentation

Algebra II Chapter 7 7-1 Roots and Radical Expressions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Algebra II Chapter 77-1 Roots and Radical Expressions MA.A.1.4.1, MA.A.1.4.4, MA.A.3.4.1

  2. Vocabulary • N th root : for any real numbers a and b, and any positive integer n, if an = b, then a is an nthroot of b • Radicand : the number under the radical sign. Example for √2, then 2 is the radicand • Index : gives the degree of the root. Example √3 has an index of 2, and ∛8 has an index of 3 • Principle Root : when a number has 2 real roots, the positive root is called the principle root. Example: √4 has both 2 and -2 as roots; 2 is the principle root

  3. Vocabulary • Nth root of an, a < 0 : for any negative real number a, n√(an) = |a| when n is even • Simplify • √(4x6) = √(2²(x³)²) = 2|x³| • 3√(a³b6) = 3√(a³(b²)3 = 3√(ab²)3 = ab²

  4. Find all real roots • 1) find the cubed root of 0.008, -1000, and 1/27 • Since (0.2)³ = 0.008, then 0.2 is the cube root of 0.008 • Since (-10)³ = -1000, then -10 is the cubed root of -1000 • Since (1/3)³ = 1/27, then 1/3 is the cube root of 1/27 • 2) find the 4th roots of 1, and 16/81 • Since 14and (-1)4 = 1 then 1 and -1 are 4th roots of 1 • Since (2/3)4 = 16/81 and (-2/3)4 = 16/81, then 2/3 and -2/3 are 4th roots of 16/81

  5. Algebra II Chapter 77-2 Multiplying and Dividing Radical Expressions MA.A.2.4.2, MA.A.3.4.1, MA.A.3.4.3

  6. Vocabulary • Rationalize the Denominator : rewrite it so there are no radicals in any denominator, and no denominator in any radical • Example : √(2/5) or √2/√3

  7. Multiplying Radical Expressions • Property : if n√a and n√b are real numbers, then n√a · n√b = n√(ab) • Multiply and simplify if possible • √2 · √8 = √ 16 = 4 • 3√(-5) · 3√25 = 3√(-125) = -5 • Simplifying Radical Expressions • √(72x³) = √(36·2·x²·x) = 6x√(2x) • 3√(80n5) = 3√(8·10·n³·n²) = 2n 3√(10n²)

  8. Multiplying Radical Expressions • Multiply and simplify assume all variables are positive: • ∛(54x²y³) ∙∛(5x³y⁴) = ∛(54x²y³∙ 5x³y⁴) • Factor into perfect cubes ∛(3³x³(y²)³∙10x²y) • Simplify 3xy² ∛(10x²y)

  9. Dividing Radical Expressions • Property : if n√a and n√b are real numbers and b ≠ 0, then n√a / n√b = n√(a/b) • Example: ∛32/∛(-4) = ∛(32/-4) = ∛(-8) = -2 • ∛(162x⁵)/∛(3x²) = ∛(54x³) = ∛(3³x³∙2) = 3x∛2

  10. Rationalize the Denominator • A) √2/√3  multiply both top and bottom by √3  (√2∙√3)/(√3∙√3) = √6/3 • B) √x³/√(5xy)  mult top and bottom by √(5xy) (√x³∙√(5xy))/ 5xy  √(5x4y)/(5xy) (x²√5y)/5xy (x√5y)/5y

  11. Algebra II Chapter 77-3 Binomial Radical Expressions MA.A.3.4.1, MA.A.3.4.2, MA.A.3.4.3

  12. Vocabulary • Like Radicals : radical expressions that have the same index and same radicand. To add or subtract like radicals, use the distributive property. • Example: 5 ∛x - 3∛x = (5-3)∛x = 2∛x

  13. Simplifying before adding or subtracting • Simplify 6√18 + 4√8 - 3√72 • 6√18 + 4√8 - 3√72 = 6√(3²•2) + 4√(2²•2) - 3√(6²•2) = (6•3)√2 + (4•2)√2 – (3•6)√2 = 8√2

  14. Multiplying Binomial Radical Expressions • Multiply radical expressions that are binomials, in the same way you multiply other binomials: FOIL • Example: multiply (3 + 2√5)(2 + 4√5) • 6 + 12√5 + 4√5 + 8(√5)² • 6 + 16√5 + 40 • 46 + 16√5

  15. Multiplying Conjugates • Conjugates are expressions such as (√a + √b) and (√a - √b) • (√a + √b) (√a - √b) = (√a)² - (√b)² = a – b • Multiply (2 + √3)(2 - √3) = 2² - (√3)² = 4 – 3 • = 1

  16. Rationalizing Binomial Radical Expressions • Rationalize the denominator of (3+√5)/(1- √5) - multiply the numerator and denominator by the conjugate of the denominator • (3+√5)(1+√5)/(1-√5)(1+√5) • (3 + 3√5 +√5 + 5)/(1 – 5) • (8 + 4√5)/(-4) = • -2 - √5

  17. Algebra II Chapter 77-4 Rational Exponents MA.A.1.4.4, MA.A.2.4.2, MA.A.3.4.2

  18. Vocabulary • Rational Exponent : using a fractional exponent to represent a radical. • Example: √25 = 251/2 • ∛27 = 271/3 • ⁴√16 = 161/4 • Rational Exponent : if the nth root of a is a real number and m is an integer, then • a1/n = n√a and am/n = n√(am) = (n√a)m

  19. Properties of Rational Exponents

  20. Simplifying Numbers with Rational Exponents • a. (-32)3/5 • Method 1: ((-2)⁵)3/5= (-2) ⁵•3/5 = (-2)³ = -8 • Method 2: (⁵√(-32))³ = (-2)³ = -8 • b. 4-3.5 = 4-7/2 • Method 1: (2²) -7/2 = 22•-7/2 = 2-7 = 1/27 = 1/128 • Method 2: 1/47/2 = 1/√(4)7 = 1/27 = 1/128

  21. Algebra II Chapter 77-5 Solving Radical Equations MA.A.3.4.1, MA.A.3.4.2

  22. Vocabulary • Radical Equation : is an equation that has a variable in the radicand or has a variable with a rational exponent. • Example: 3+√x = 10, or (x-2)2/3 = 25 • To solve a radical equation, isolate the radical on one side of the equation and then raise both sides of the equation to the same power • Example: if n√x = k; then (n√x)n = kn and x = kn

  23. Solving Radical Equations with Index 2 • Solve: 2+ √(3x-2) = 6 • Isolate the radical: √(3x-2) = 4 • Square each side: (√(3x-2))² = 4² • 3x-2 = 16 • 3x = 18 • x = 6

  24. Solving Radical Equations with Rational Exponents • Solve : 2(x-2)2/3 = 50 • Divide by 2: (x-2)2/3 = 25 • Raise each side to the inverse power (3/2) • ((x -2)2/3)3/2 = 253/2 : if you have an even root (ie denominator) then you must take absolute value of variable side • |x - 2| = ± 125 • x = 127 or x = -123

More Related