Lesson1: Perfect Squares



Unit 4: Radicals and RootsLesson1: Perfect SquaresSquared: to raise a number to the second power or have an exponent of 2Example: 4 raised to the second power is 4 squared or 42 42 = 4 x 4 = 16 122 = 12 x 12 = 144Perfect Square: Any number whose square root is a whole numberExample: 36 is a perfect square because 36 = 6 (whole number) 12 is not a perfect square because 12 =3.46 (not a whole number)First 12 perfect squares: 12 = 142 = 1672 = 49102 = 10022 = 452 = 2582 = 64112 = 12132 = 962 = 3692 = 81 122 = 144Examples and Notes from Class:Lesson 2: Perfect CubesCubed: to raise a number to the third power or have an exponent of 3Example: 6 raised to the third power is 6 cubed or 63 63 = 6 x 6 x 6 = 256 33 = 3 x 3 x 3 = 27To Write a Cube Root: 3125 this means, what number, multiplied 3 times by itself equals 125? 5 x 5 x 5 = 125. So, 3125 = 5Perfect Cube: Any number whose cube root is a whole numberExample: 64 is a perfect cube because 364 = 4 (4 x 4 x 4 = 64) 100 is not a perfect cube because 3100 = 4.64 (not a whole number)First 6 perfect Cubes:13 = 1 x 1 x 1 = 1 23 = 2 x 2 x 2 = 8 33 = 3 x 3 x 3 = 2743 = 4 x 4 x 4 = 6453 = 5 x 5 x 5 = 12563 = 6 x 6 x 6 = 256Notes from Class: Lesson 3: Simplify Roots (Radicals)Radical: another name for the square root symbol Evaluate Square Roots64 = 8 because 8 x 8 = 6464 = -8 because -8 x -8 = 64So, we write: 64 = ±8 (this is read, the square root of 8 equals plus or minus 8)Principal Root: the positive solution to a square root problemThe principal root of 63 is +8Evaluate Cube Roots3256 = 6 because 6 x 6 x 6 = 2563256 ≠ (does not equal) - 6 because -6 x -6 x -6 = -25644767533591500Example Evaluate and Order Roots: 81, 364 , 4 , 3256Evaluate: Order from Smallest to Largest: 2305050-254000Notes and Examples from ClassLesson 4: Identify Irrational NumbersRational Number: can be expressed as the ratio of 2 integers (fraction) As a decimal, a rational number terminates OR repeatsExamples: 8 is rational because it can be expressed as 81-34 is rational because it can be expressed as -341 .35 is rational because it can be expressed as 35100 AND because it terminates .124124124… is rational because it is a repeating decimalIrrational Number: cannot be expressed as the ratio of 2 numbers (fraction) As a decimal it does NOT terminate OR RepeatExamples: .543823… -63.847362514… π (= 3.141592…)Example: 400 = ±20Example: 48 = 6.928203…Expressions: Rational or Irrational?Add, Subtract, Multiply or Divide 2 Rational Numbers: answer is Rational Example: 36 ÷ 5 36 is rational and 5 is rational so the answer is rationalAdd, Subtract, Multiply or Divide an Irrational Number and a Rational Number: answer is Irrational Example: 40 ÷ 5 40 is irrational and 5 is rational so the answer is irrationalNotes and Examples from ClassLesson 6: Simplify RadicalsPrime Numbers: any number whose ONLY factors are 1 and itselfExample: 2 (because the only numbers you can multiply to get 2 are 2 and 1) 13 (because the only numbers you can multiply to get 13 are 13 and 1)Prime Factorization: break down a number into its prime factors270510018732500 40 = 2 x 2 x 2 x 5 Create a Factor Tree to Prime Factor the number 40-20002550038000Simplify Square Roots: Example 1: Simplify 24-26670025844500 Prime Factor 24 = 26 Take out 1 of any pair-20002523812500Example 2: Prime Factor 180-20002510922000 = 65 Take out 1 of any pair Simplify Cube RootsExample: 3120Prime Factor 120 using a Factor Tree: 2733675317500-13335060960000457200285750003120 = Rewrite the problem Take out 1 of any triplet of the same number 3120 = 2315Notes and Examples from Class: Lesson 7: Multiply and Divide Radical in an ExpressionCoefficient: the number in front of the radicalRadicand: the number inside the radicalExample: 57 5 is the coefficient and 7 is the radicand66675048895000Multiply Radicals: Multiply the coefficients and the radicands separately, then simplifyExample: -200025562610004 x 2 = 8Multiply the coefficients8 x 3 = 24Multiply the radicandsSimplify: -10477518224500Examples from Class:80962531432500Dividing Radicals: Divide the coefficients and radicands separately, then simplify-4762546164500Example: Divide 32 by 8 and 24 by 2035433000 Simplify 12-4762516319500 SimplifyMultiplying Radicals and the Distributive PropertyNotes and Examples from Class:Lessons 8: Add and Subtract RadicalsRadicands (the number under the radical sign) MUST be the same in order to add or subtract12382562928500Examples: If the radicands aren’t the same, simplify to see if you can make them the same62865024574500Example: 518 + 250 Simplify80010024701500 Add203835027432000023622000Combine Like Terms to Add or Subtract: (Terms with the same Radicand) Notes and Examples from Class: Lessons 9: Rationalize the Denominator of an Expression504825109347000Simplified Fraction: cannot have a radical in the denominatorExamples of Removing the Radical in the DenominatorExample 1: Find the square root of the denominatorExample 2: 024701500 Multiply the top and bottom times the square root 6 ÷ 6 = 1, so this will not change the value of the fractionThe final, simplified answer may have a radical in the numerator, but not the denominator0101917500Example 3: The 10 on the top and bottom cancel each other out Notes and Examples from Class ................
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