Lesson 4 β’ 40 min read
Radicals
In This Lesson
1Radical Notation
βΏβa = a^(1/n)
n
Index (root)
β
Radical sign
a
Radicand
Common Radicals
βa = Β²βa = a^(1/2) β Square root
Β³βa = a^(1/3) β Cube root
β΄βa = a^(1/4) β Fourth root
Perfect Squares
1, 4, 9, 16, 25, 36, 49, 64, 81, 100...
β4 = 2, β9 = 3, β16 = 4, β25 = 5
Important Notes
- β’ βa is only real if a β₯ 0 (for even roots)
- β’ Β³βa is real for all values of a (odd roots)
- β’ βaΒ² = |a| (absolute value, always positive)
- β’ When index is not written, it's assumed to be 2
2Laws of Radicals
Product Rule
β(ab) = βa Γ βb
The root of a product equals the product of the roots
β12 = β(4 Γ 3) = β4 Γ β3 = 2β3
Quotient Rule
β(a/b) = βa / βb
The root of a quotient equals the quotient of the roots
β(16/9) = β16 / β9 = 4/3
Power Rule
(βa)βΏ = β(aβΏ) = a^(n/2)
A root raised to a power or a power under a root
(β5)β΄ = 5Β² = 25
Index Rule
α΅β(βΏβa) = α΅βΏβa
A root of a root multiplies the indices
Β³β(β64) = βΆβ64 = 2
3Simplifying Radicals
Steps to Simplify
- Find the largest perfect square factor of the radicand
- Rewrite as product of perfect square and remaining factor
- Take the square root of the perfect square
- Write result as coefficient times remaining radical
Example 1: Simplify β72
β72 = β(36 Γ 2) β 36 is largest perfect square factor
= β36 Γ β2
= 6β2
Example 2: Simplify β50
β50 = β(25 Γ 2)
= 5β2
Example 3: Simplify β48xΒ³
β48xΒ³ = β(16 Γ 3 Γ xΒ² Γ x)
= β16 Γ βxΒ² Γ β(3x)
= 4xβ(3x)
Example 4: Simplify Β³β54
Β³β54 = Β³β(27 Γ 2) β 27 = 3Β³ is perfect cube
= 3Β³β2
4Operations with Radicals
Addition & Subtraction
Only combine LIKE RADICALS
Like radicals have the same index and radicand
Example: 3β2 + 5β2
= (3 + 5)β2 = 8β2
Example: 7β3 - 2β3
= 5β3
Example: β12 + β27 (simplify first!)
= β(4Γ3) + β(9Γ3)
= 2β3 + 3β3
= 5β3
Multiplication
Example: β3 Γ β12
= β(3 Γ 12) = β36
= 6
Example: 2β5 Γ 3β2
= (2 Γ 3) Γ β(5 Γ 2)
= 6β10
Example: β2(β6 + β8)
= β2 Γ β6 + β2 Γ β8
= β12 + β16
= 2β3 + 4
= 4 + 2β3
Division
Example: β20 / β5
= β(20/5) = β4
= 2
5Rationalizing Denominators
Rationalizing means eliminating radicals from the denominator. We do this by multiplying by a clever form of 1.
Single Term Denominator
Multiply by βa/βa
Example: 5/β3
= (5/β3) Γ (β3/β3)
= 5β3 / (β3 Γ β3)
= 5β3 / 3
= (5β3)/3
With Coefficients
Example: 6/β8
First simplify β8 = 2β2
= 6/(2β2) = 3/β2
= (3/β2) Γ (β2/β2)
= (3β2)/2
Binomial Denominator (Conjugates)
Multiply by the conjugate: (a + βb) and (a - βb) are conjugates
Example: 2/(3 + β5)
= [2/(3 + β5)] Γ [(3 - β5)/(3 - β5)]
= 2(3 - β5) / [(3)Β² - (β5)Β²]
= (6 - 2β5) / (9 - 5)
= (6 - 2β5)/4 = (3 - β5)/2
Why Rationalize?
- β’ Standard mathematical form (easier to read and compare)
- β’ Easier to perform further calculations
- β’ Required form on many standardized tests
- β’ Conjugate method uses: (a+b)(a-b) = aΒ² - bΒ²