Algebra II Chapter 77-1 Roots and Radical

ExpressionsMA.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 an = b, then a is an nth root 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

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

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 14 and (-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

Algebra II Chapter 77-2 Multiplying and Dividing

Radical ExpressionsMA.A.2.4.2, MA.A.3.4.1,

MA.A.3.4.3

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

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)

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)

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

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

Algebra II Chapter 77-3 Binomial Radical Expressions

MA.A.3.4.1, MA.A.3.4.2, MA.A.3.4.3

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

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

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

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

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

Algebra II Chapter 77-4 Rational Exponents

MA.A.1.4.4, MA.A.2.4.2, MA.A.3.4.2

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

Properties of Rational Exponents

Property Example

am•an = am+n 8⅓•8⅔ = 8⅓+⅔ = 8¹ = 8

(am)n = amn (5½)4 = 5½•4 = 5² =25

(ab)m = ambm (4•5)½ = 4½ • 5½ = 2• 5½

a-m = 1/am 9-½ =1/9½ = 1/3

am/an = am-n π3/2/π½ = π3/2- ½ =π¹ = π

(a/b)m = am/bm (5/27)⅓ = 5⅓/27⅓ = 5⅓/3

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

Algebra II Chapter 77-5 Solving Radical Equations

MA.A.3.4.1, MA.A.3.4.2

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

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

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