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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

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Algebra II Chapter 7 7-1 Roots and Radical Expressions


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    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