Download Algebra II Chapter 7 7-1 Roots and Radical

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 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 7
7-2 Multiplying and Dividing
Radical Expressions
MA.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 7
7-3 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 7
7-4 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 7
7-5 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
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•
•
•
•
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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
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•
•
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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