NAME UNIT 4 ALGEBRA II. NOTES PACKET ON RADICALS, RATIONALS d COMPLEX NUMBERS

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1 NAME UNIT 4 ALGEBRA II NOTES PACKET ON RADICALS, RATIONALS d COMPLEX NUMBERS

2 Properties for Algebra II Name: PROPERTIES OF EQUALITY EXAMPLE/MEANING Reflexive a - a Any quantity is equal to itself. Symmetric If a = b, then Z? = a. If the first quantity equals a second,quantity, then the second quantity also equals the first. Transitive If a = b and b ~ c, then a = c. If the first quantity equals the second quantity and the second quantity equals the tliird Quantity, then the first and third quantities are equal. Addition Ua - b, then a + c = b + c. You can add the same number to each side of an equation. Multiplication If a - b and c = 0, then ac - be. You can multiply each side of an equation by the same nonzero number. Substitution If a = b -»- x and a + b = 0, then a + b = (b -h.r) 4- b =,r + 2b = 0. You can replace an expression with another expression or real number that has the same value. PROPERTIES OF REAL NUMBERS EXAMPLE/MEANING Commutative of addition For any numbers a and b, a + b = b + a. You can change the order of the numbers and the answer is not affected. Commutative of multiplication For any numbers a and b, ab = ba. You can change the order of the numbers and the answer is not affected. Associative of addition For any numbers a, b, and c, (a + b) -f c = a + (b + c). You can regroup the numbers and it does not affect the sum. Associative of multiplication For any numbers n, b, and c, (a V) c = a (b c). You can regroup the numbers and it does not affect the product.

3 Identity of addition For any number a, a i- 0 = a, or 0 + a - a. The sum of zero and a number is always that number. Identity of multiplication For any number a, a 1 = a, or 1 a = a. The product of 1 and any other factor is always that factor. Additive inverse (Inverse Property of Addition) Multiplicative inverse (Inverse Property of Multiplication) Distributive (of mujtiplication with respect to addition) For every number a, a 4- (~a) = 0. The sum of a number and its opposite is zero. For every nonzero r, where a ~ 0 and h j- n - - = i»t-u,b a ~ i. The product of a number and its reciprocal isl For any numbers a, b, and c, a(b + c) = ab + ac, and (b 4- c)a - ba 4- ca. A number outside the parentheses can be used to multiply each term within the parentheses. PS-

4 ROOTS OF REAL NUMBERS Algebra II notes DEFINITION OF SQUARE ROOT: For any real numbers a and b, if a2 = b, then a is a square root of b. DEFINITION OF NTH ROOT: For any real numbers a and b, and any positive integer n, if an = b, then a is an nth root of b. EXAMPLES Find each root. 1) ±V49*2 2) -V(a2 + I)4 3) V32*10?15 4) x* 5) 243 6) -3125

5 RADICAL EXPRESSIONS Algebra II notes EXAMPLES Simplify. 1) V80 2) 4V54 3) 4) V54 5) 6) 7) V81m4n5 8) 9) 10)

6 13) 2V2-2V2 + 3V3 14) 2V20-2V25 + 3V80 15) 2V36-2V48 + 3V24 16) (3V5-4)(3V5 + 4) 17) (2V6-3V2)(3 18) (5 + V8)(4 - P9-'

7 RATIONAL EXPONENT Algebra II notes Evaluate each expression in your calculator. -1 1) 81T 2) 32~s 3) if 42 DEFINITION OF RATIONAL EXPONENTS: For any nonzero real number b, and any integers /Hand /?, with n > 1, EXAMPLES Write the expression in simplest radical form. 2 5 J_ 2 4) AT 3 x 3 5) c 4 e 5 6) ov ? / r\ 7) *2ysZ6 8) j 9)

8 Write the expression with rational exponents. 10) V 11) 12) 3 X 13) 14),10 15) V^8 y6.1

9 Algebra II Name: _ Notes - Complex Numbers In math, the letter / is defined as the "imaginary unit" where / = v-1. Knowing this, what is /2? 1. Patterns in powers of i You can use these patterns to help simplify radical expressions. Simplify FIRST, then add, subtract, multiply, or divide. Fill in the pattern for /. i2 = is i1 I = What is a method for simplifying / raised to an exponent? Example 1 : Simplify i25 You Try: Simplify 3" / = 1 /'

10 2. You can use this to simplify. For a complex expression to be simplified, you must: Never leave a negative under a J~. Never leave / with an exponent. Never leave a V m the denominator. Never leave / in the denominator. Follow all other rules for simplifying. Example 2: Simplify V-i You Try: Simplify V- 54 2VJ 2/V3 Example 3: You Try: Simplify 2V-98;t5 Example 4: Simplify V-27 «4V-21 You Try: Simplify - 2V- 72 6V- 30 Example 5: 1-8 Simplify J You Try: Simplify -12 ~7T 2/V2 2/V2 V5 2/VTo Remember: V-1 = /, but V-1 = -1. So, just because a radicand is negative doesn't mean there will be

11 3. Performing operations with complex numbers. When adding, subtracting, or multiplying complex numbers, just treat the / like a variable, then simplify using the patterns of powers of/. Example 6: Simplify (2 + 3/) - (5-3z) 2 + 3/-S + 3/ You Try: Simplify (3-2i) - (8 + Si) Example 7: Simplify (4 + 2i)+ 3i(6-5/) You Try: Simplify (3-5i) + 5/3(2-3/) 4 + 2/ + 18/ / Example 8: Simplify (l 5 + 3/X2-4/) You Try: Simplify (3-2/)2 15 3z -Ai 30-60/ +6i -12/ z-60/-12/ /-12(-l) 30-54z J 4. Conjugates - the conjugate of a + b/ is a - b/. The conjugate of 3-5/ is 3 + 5/. The conjugate of 4 + Hs4-/. You Try: What is the conjugate of /? You Try: Simplify (2 + 3/X2-3/) When you multiply conjugates, the result is always what classification of number?

12 Classifying Numbers Name: Complex Numbers Real numbers 1 +2i n -f Rational numbers -'""" Integers J_ 6 Pure Imaginary 0 + 2i Irrational numbers Class Natural Number Whole Number Integer Rational Number Irrat onal Number Real Number Pure Imaginary Number Complex Number Symbol N Z Q R C Description Natural numbers are defined as positive counting numbers: N = {1,2,3,4,...}. Whole numbers are defined as non-negative counting numbers: -[0, 1, 2, 3, 4,... }. Integers extend N by including the negative of counting numbers: Z = {...,-4, -3, -2, -1,0, 1,2,3,4,... }. The symbol Z stands for Zahlen, the German word for "numbers". A rational number is the ratio or quotient of an integer and another non-zero integer: E.g.: -100, -201/4, -1.5, 0, 1, 1.5, V/2 2%, 1.75, 0.3 Irrational numbers are numbers which cannot be represented as fractions. E.g.: V2, \/3, n, e Real numbers are all numbers on a number line. The set of K is the union of all rational numbers and all irrational numbers. An imaginary number is a number which square is a negative real number, and is denoted by the symbol /, so that i2 = -1. E.g.: -5/, 3/, 7. Si A complex number consists of two part, real number and imaginary number, and is also expressed in the form a + b/" (/' is notation for imaginary part of the number). E.g.: , 2 + O/, 0-2/ [y. _. _. LM«