CEE 200
Math Plus
Juvywen Pollentes Española, ME-CpE
Course Teacher
Book Reference
1. College Algebra and Trigonometry
• by Barnett
2. Algebra and Trigonometry
• by James Stewart
ALGEBRA
Subtopics
1.Real Numbers and Properties of Real
Numbers
2. The real Number Lines
3. Addition and Multiplication of Real
Real Numbers
4. Further Operations and Properties
5. Index Law
6. Polynomials
7. Special Products and Factoring
8. System of Linear Equation
9. Quadratic Equation
10. Imaginary Numbers
11. Worded Problem
Rational Exponents
• For n a natural number and b a real number, the principal nth root
𝒏
of b, denoted by b1/n or 𝒃, is
1. The real nth root of b if there is only one.
2. The positive nth root of b if there are two real nth roots
3. Undefined if b has no real nth root.
• Principal 𝑛th Root:
• Given 𝑛 is a natural number (positive integer) and 𝑏 is a real number,
the principal 𝑛th root of 𝑏 is defined as follows:
1. The Real 𝑛th Root of 𝑏 If There Is Only One:
Example:If 𝑛=3 and 𝑏=8, then 8^1/3=2. The cube root of 8 is 2, and
since it's the only real cube root, this is the principal root. Similarly, if
𝑛=3 and 𝑏=−8, then (−8)^1/3=−2. The cube root of -8 is -2, which is the
only real root.
2. The Positive 𝑛th Root of 𝑏 If There Are Two Real 𝑛th Roots:
• Example: If 𝑛=2 (square root) and 𝑏=9, then 9^1/2=3. There are two
square roots of 9: 3 and −3 (since 3^2=9 (−3)^2= 9). The principal
square root is the positive one, which is 3.
3. Undefined If b Has No Real nth Root:
• Example:
• If n=2 and b=−9, then the expression (−9)^1/2 is undefined in the real
number system because no real number squared gives a negative result.
Hence, the square root of a negative number does not exist in the real
numbers.
𝑛
• In the notation 𝑏, the symbol
is called a radical, n is
called the index, and b is the radicand.
• Evaluate each expression
𝟏
𝟐
• 𝟏. 𝟗
• 2. 𝟏𝟐𝟏
𝟑
• 3. −𝟏𝟐𝟓
• 4. −𝟏𝟔
• 𝟓. 𝟐𝟕
𝟏
𝟑
𝟏
𝟒
• 1.
𝟒
𝟖𝟏𝒙𝟖
2.
𝟑
−𝟏𝟐𝟓𝒙𝟑 𝒚𝟔
3. −𝟔𝟒 𝒙𝟐
Rational Exponent and Radicals
Properties of Radicals
For n a natural number greater than 1, and x and y are positive real
numbers:
𝐧
𝐧
𝟑
𝐧
𝐱 =𝐱
𝐱𝐲 =
𝐧
𝐧
𝐱 𝐲
𝟓
𝐱𝟑 = 𝐱
𝐱𝐲 =
𝟓
𝟓
𝐱 𝐲
Properties of Radicals
For n a natural number greater than 1, and x and y are
positive real numbers:
𝐧
𝐧
𝐱
𝐱
=𝐧
𝐲
𝐲
𝟒
𝟒
𝐱
𝐱
=𝟒
𝐲
𝐲
1. Property 1
𝟑
•
a.)
−𝟖 . 𝟐𝟕
𝟒
• 2. Property 2
•
a)
𝟒
𝟏𝟔
𝟖𝟏
𝟏𝟔𝟐𝒙𝟒
𝟏𝟔
b.)
• 3. Property 3
• a)
𝟑
𝟕𝟐𝟗
b.)
𝟑 𝟒
• 4. Property 4
• a)
𝟓
𝒙𝟏𝟓
b.) 𝟒𝒙𝟒
• For odd and even
𝟒
b.) 𝟑𝟐 . 𝟖𝟏
𝒙𝟑𝒙𝟐𝟑
Rationalizing the Denominator; Standard Form
▪ It is often useful to eliminate the radical in a denominator by multiplying both numerator
and denominator by an appropriate expression
▪ If the denominator is of the form 𝑎, we multiply numerator and denominator by 𝑎.
• Case 1
• Case 2
• Case 3
•
𝑛
𝑎𝑚
𝑛
𝑎𝑛−𝑚 =
𝑛
𝑎𝑚+𝑛−𝑚 =
𝑛
𝑎𝑛 = a
rationalization using special products
Example
• 1
• Using the formula ( a,b)
𝑛
Case 2
𝑎𝑚
𝑛
𝑎𝑛−𝑚
• 2. Rationalized the expression
• (a)
1
𝑥+1
=
𝑛
𝑎𝑚+𝑛−𝑚 =
𝑛
𝑎𝑛 = a
Example
• 3. Case 3
•
𝟑 𝟐+𝟏
𝟑 𝟐−𝟏
Rationalizing Denominator
To rationalize the denominator, we multiply the numerator and
denominator by a suitable factor that will leave the denominator free of
radicals.
Write in simplified radical form.
𝟏𝟐𝒙𝟓 𝒚𝟐
𝟔
𝟏𝟔𝒙𝟒 𝒚𝟐
𝟔
𝟐𝒙
𝟑
𝟖𝒙𝟒
𝒚
Combination of Radicals
I. Removal of factors from radicand
II. In a sum, two or more terms involving the same radical as a factor may be
combined by factoring.
III. The product or quotient of two radicals of the same order can be
expresses as a single radicals.
IV. Some powers of a radical can be simplified by use of (𝒏 𝒂)𝒏
Example
1. Combine the following Radicals
• a.) 5 3𝑥 + 2 3𝑥
• b.) 20 + 2 45
• 2. Product and Quotient of the Radical
• a.)(2 3)(5 6)
3
3
• b.) 2𝑎 . 20𝑎2 𝑏 2
• c.) (2 3 + 3 2𝑥)(3 3 − 2 2𝑥)
• c.)
8𝑥 2
4𝑥