TOPIC 0-FUNDAMENTAL CONCEPTS OF ALGEBRA (MAT0114) REAL NUMBERS SYSTEM EXPONENTS: LAW OF INDICES ALGEBRAIC EXPRESSION RADICALS

1. REAL NUMBER SYSTEM Real Rational Integer Negative Zero Positive Natural Non integer Terminate Decimal Decimal repeat in cycle Irrational Decimal does not repeat in cycle Example1: Classify the numbers into their type (Natural, Integer, Rational, Real). 2.33, −1.5, 7 4 , 𝜋, 6 , − 11 2 , 2

INTERVALS Open interval : , Close interval: , Union: take all the values in the interval. −2,3 ∪ 2,10 = Intersection: take the intersection values only. 3 ,12)∩ −2,8 =

2. EXPONENTS & RATIONAL EXPONENT LAW OF EXPONENT LAW OF RATIONAL EXPONENT 1) 𝑎 𝑚 × 𝑎 𝑛 = 𝑎 𝑚+𝑛 2) 𝑎 𝑚 ÷ 𝑎 𝑛 = 𝑎 𝑚−𝑛 3) 𝑎 𝑚 𝑛 = 𝑎 𝑚𝑛 4) 𝑎𝑏 𝑛 = 𝑎 𝑛 𝑏 𝑛 5) 𝑎 𝑏 𝑛 = 𝑎 𝑛 𝑏 𝑛 1) 𝑎 1 𝑛 = 𝑛 𝑎 2) 𝑎 𝑚 𝑛 = 𝑛 𝑎 𝑚 = 𝑛 𝑎 𝑚 3) 𝑎 𝑚 𝑛 = 𝑎 1 𝑛 𝑚 = 𝑎 𝑚 1 𝑛 THEOREM ON NEGATIVE EXPONENTS 1) 𝑎 −𝑛 = 1 𝑎 𝑛 2) 𝑎 −𝑚 𝑏 −𝑛 = 𝑏 𝑛 𝑎 𝑚 3) 𝑎 𝑏 −𝑛 = 𝑏 𝑎 𝑛

3. Algebraic expression Algebraic expression is obtained by applying additions, subtractions, multiplications, divisions, powers or taking roots to collection of variables and real numbers. Simplify the algebraic expression: a) 2𝑢+3 𝑢−4 +4𝑢 𝑢−2 =

PRODUCT FORMULAS 1) 𝑥+𝑦 𝑥−𝑦 = 𝑥 2 − 𝑦 2 2) 𝑥+𝑦 2 = 𝑥 2 +2𝑥𝑦+ 𝑦 2 1) 𝑥+𝑦 𝑥−𝑦 = 𝑥 2 − 𝑦 2 2) 𝑥+𝑦 2 = 𝑥 2 +2𝑥𝑦+ 𝑦 2 3) 𝑥−𝑦 2 = 𝑥 2 −2𝑥𝑦+ 𝑦 2 4) 𝑥+𝑦 3 = 𝑥 3 +3 𝑥 2 𝑦+3𝑥 𝑦 2 + 𝑦 3 5) 𝑥−𝑦 3 = 𝑥 3 −3 𝑥 2 𝑦+3𝑥 𝑦 2 − 𝑦 3 EXERCISES: Find the product i) 2 𝑟 2 − 𝑠 2 𝑟 2 + 𝑠 = ii) 2𝑥+3𝑦 3 =

FACTORING FORMULAS 1) Difference of two squares: 𝑥 2 − 𝑦 2 = 𝑥+𝑦 𝑥−𝑦 2) Difference of two cubes: 𝑥 3 − 𝑦 3 = 𝑥−𝑦 𝑥 2 +𝑥𝑦+ 𝑦 2 3) Sum of two cubes: 𝑥 3 + 𝑦 3 = 𝑥+𝑦 𝑥 2 −𝑥𝑦+ 𝑦 2 EXERCISES: Factor each polynomials i) 36 𝑟 2 −25 𝑡 2 = ii) 125 𝑥 3 −8=

FACTORING BY TRIAL & ERROR Trying various possibilities to factor the algebraic expression. Factorize: i) 6 𝑥 2 −7𝑥−3=

FACTORING BY GROUPING If a sum contains four or more terms, it may be possible to group the terms in a suitable manner then find a factorization by using distributive properties. Factor each expression: i) 3 𝑥 3 +3 𝑥 2 −27𝑥−27=

ABSOLUTE VALUE - is defined as the distance from the origin to the specific point/value on real number line. i) if 𝑎≥0, 𝑡ℎ𝑒𝑛 𝑎 =𝑎 ii) if 𝑎<0, 𝑡ℎ𝑒𝑛 𝑎 =−(𝑎) Example 1: Evaluate a) 7−12 = b) 𝜋−6 =

4.0 RADICALS PROPERTIES OF nth ROOT 1) 𝑛 𝑎𝑏 = 𝑛 𝑎 𝑛 𝑏 1) 𝑛 𝑎𝑏 = 𝑛 𝑎 𝑛 𝑏 2) 𝑛 𝑎 𝑏 = 𝑛 𝑎 𝑛 𝑏 3) 𝑚 𝑛 𝑎 = 𝑚𝑛 𝑎 4) 𝑛 𝑎 𝑛 = 𝑎 if n is even 5) 𝑛 𝑎 𝑛 =𝑎 if n is odd

RATIONALIZE DENOMINATOR OF QUOTIENTS FACTOR IN DENOMINATOR MULTIPLY NUMERATOR AND DENOMINATOR BY: RESULTING FACTOR: 𝑎 𝑎 2 =𝑎 3 𝑎 3 𝑎 2 3 𝑎 3 =𝑎 7 𝑎 3 7 𝑎 4 7 𝑎 7 =𝑎 Examples: Simplify and rationalize the denominator a) 1 5 b) 2 𝑥 c) 3 2 𝑥 4 𝑦 4 9𝑥