Never leave a NEGATIVE EXPONENT or a ZERO EXPONENT in an answer in simplest form!!!!!


 Marsha Lester
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1 1 ICM Unit 0 Algebra Rules Lesson 1 Rules of Exponents RULE EXAMPLE EXPLANANTION a m a n = a m+n A) x x 6 = B) x 4 y 8 x 3 yz = When multiplying with like bases, keep the base and add the exponents. a m a n = am n OR an a m = 1 a m n m > n A) x8 x 3 = B) x y 5 x 7 y 6 = When dividing with like bases, keep the base and subtract the exponents. (a m ) n = a mn A) (x 5 ) 3 = Power to a Power keep the base and multiply the exponents. (ab) m = a m b m A) (x 5 y 3 ) 3 = B) (x 3 z 4 ) 4 = Power to a Product Raise everything in the parentheses to the power. ( a b ) m x A) ( = am y 4) = b m B) ( 3x4 y 5 4x y 7)3 = Power to a Quotient Raise everything in the parentheses to the power. a m = 1 a m OR 1 a m = am A) x y 3 z 5 x 6 = B) x 4 3y = Change a negative exponent to a positive exponent by moving the base to either the denominator or the numerator of a fraction. b 0 = 1 A) (x 4 ) 0 = B) 3x 0 y 5 = Any base raised to the zero power equals 1. Never leave a NEGATIVE EXPONENT or a ZERO EXPONENT in an answer in simplest form!!!!!
2 Examples: Simplify 1. (x 3 y )(x 4 y 4 z). (3x 3 y)( 5x y ) 3. x4 y 5 x 3 y 4. (x 3 y 4 ) (x 6 )(y 3 ) 5. ( 3a b c 4 4a 6 b 4 c 8) 6. ( x 6 )( x 3 ) (x 5 ) 3 8. (3a b 5 ) (a b) 4 3(ab 4 ) ( 6a 4 ) (3a 6 b 3 ) 7. ( 3a3 b ) 3
3 3 ICM Unit 0 Algebra Review Lesson 1 Rational (Fractional) Exponents Rule for Converting to Rational & Radical Notation x a b b = x a Write each expression in simplest radical form: x 1 8. y 1 9. a 10. (9a) (16x 5 ) Write each expression in exponential form: x (3x) 7
4 4 Write each expression in simplest radical form leave only one radical sign in your answer: 1. x x 3 3. (5x) y 7 y b x 9 y x a 3 b (x 1 ) x 1 ( x 1 + x 3 ) 3. (3x 10 ) 1 5
5 5 ICM Unit 0 Algebra Review SHOW ALL WORK Lesson 1 Homework Let a and b be real numbers and let m and n be integers. Product of Powers Property a m a n = a m+n Negative Exponent Property a m = 1 a m Power of a Power Property (a m ) n = a mn Zero Exponent Property a 0 = 1 Power of a Product Property (ab) m = a m b m Rational Exponent Property a m n n = (a m ) Evaluate the expression. Quotient of Powers Property = a n am n a 0 n = ( a) m Power of a Quotient ( a b )m = am an b 0 a m (5 ) 3 4. ( 3 7 )3 6. ( 9)( 9) 3 Simplify the expression. 7. a 6 a ( 3 x ) 5 1. (x 3 ) 0 8. (x 5 ) 15. (x 4 y 7 ) 3. (10x 5 y 3 ) 3 9. (4a b 3 ) x11 y 10 x 3 y 1 3. x 1 y xy 10. x8 x x 4 y 0 4. (4x y 5 ) 11. x5 x x3 y 9 0x y 5. x y 6xy 1 1. x6 x x5 x 6. xy9 7y 3y 1x ( 4a3 b 4) 0. x5 y x 4 y xy 7x5 y 7x 4 4y
6 6 ICM Unit 0 Review of Algebra Lesson Factoring I. Greatest Common Factor (GCF) if possible, always do this FIRST. A. 4a b 18ab B. 5x y 0xy z + 35y 3 z C. x 3 yz 3 7xy 5 z II. Difference of Squares Factoring a b = (a b)(a + b) *** Always check for a GCF first!!!! A. x 9 B. x 49 C. x 36y D. 16x 1 E. x + 5 F. 1 + x G. 4x 5 54xy 6 H. 4x 64 I. x 4 16 III. Sum and Difference of Cubes SUM a 3 + b 3 = (a + b)(a ab + b ) DIFFERENCE a 3 b 3 = (a b)(a + ab + b ) *** Always check for a GCF first!!!! A. x 3 15 B. 7x 3 1 C. 64x 3 8y 3 D. 3x E. 3x y 3 F. x G. 3x 3 4y 3 H. 448x I. 8x 6 + 7y 3
7 7 IV. Factoring Trinomials x + bx + c *** Always check for a GCF first!!!! A. x + 9x + 0 B. x 7x + 10 C. x + 3x 40 D. x 3x 10 E. x 8x 90 F. x 4 7x + 1 V. Factoring Trinomials a x + bx + c *** Always check for a GCF first!!!! A. x + 7x + 6 B. x 9x + 4 C. 3x + 5x + D. 6x 4x 4 E. 6x + 11xy + 4y F. 5x 4 17x + 14 VI. Factoring by Grouping 1. Check for GCF. Group 3. GCF of each group 4. Binomial GCF. A. x 3 5x + 3x 15 B. 4x + 0x 3xy 15y C. 3x 3 6x + 15x 30 D. x + ab ax bx E. x 3 + x 9x 18 F. 9x x 4x 16
8 8 ICM Unit 0 Algebra Review Lesson 3 Simplifying Rational Expressions Rational Function: Quotient of polynomials written in simplest form with a denominator that cannot be equal to zero (Ex: x+ (x 1)(x+3) ; x 1 or 3) A rational function is undefined when the denominator is equal to zero. 1) Factor the denominator completely. ) Set each factor equal to zero and solve. 3) State what values make the function undefined. Example: For what values of x is the following function undefined? f(x) = x 3x+ 1) Factor the denominator completely: ) Set each factor equal to zero and solve: 3) State the undefined values: The function f(x) will be undefined at x = **From this point on, we will assume that the replacement set of the variables in the fraction includes no numbers for which the denominator will be equal to zero. x 3 4x Simplest Form: Answers to all fraction problems should be in simplest form. To put rational expressions in simplest form, factor everything completely (numerator and denominator) and then divide out (cross out) the common factors in the numerator and denominator. Simplify: 1. x +4x x 16. x 9 (x+3) 3. 3x3 x 4 x 3 6x 4. x 4x 5 x 5x 6 5. x5 3x 4 4x 3 x 3 6x +8x 6. x 8x+16 4 x
9 9 Adding and Subtracting Rational Expressions Simplify: x + 4y 1y 6x x+7 x+5 x y y 9 + y y +3y 5. x x 5 x x 10x+5 6. x 8 x+6 x 6x+8 x +x 6
10 10 Multiplying and Dividing Rational Expressions Simplify: 1. 4a 5b 15b 16a. 7a 63b3 9b 35a 3. 4x y xy 15a 3 b 3 5ab 3 x+ 4. x +x 1 x+3 x 9 5. x 3x 4 x 3 x +x x 8 6. x x 5 x +5x x x Multiplying Polynomials Simplify: 1. (3x 4 y + 4xy 3 1x 5y + 4)(x yz ). (x y 3 z 3 )( xz 8 xy 4 z )
11 11 ICM Unit 0 Algebra Review SHOW ALL WORK Lesson 3 Homework Simplify 4x 56x 6 y y x 6x 8 5x x x 3x 40 x 15 Multiply or divide (remember to factor when necessary). 4. 6x 18 x x 5x 4 x x 4x 3 5. x x 1 x x 0 4x 1 3x x 45x x 7. 9x 6x 4 5x 5 x 9x 0 3x x 14 1x 4x x x 1 5x 0 x 3x 8 x x 7x 8 x 3x 4 x 6 4x xy 35x y 14xy 10x y 4 3 Add or Subtract x x 30 x x x 3 y 3 y y 8x x x x 7 4x y 8 y 3y 11x 4
12 ICM Unit 0 Algebra Review Lesson 4 Dividing Polynomials 1
13 13
14 14 ICM Unit 0 Algebra Review SHOW ALL WORK Lesson 4 Homework
15 ICM Unit 0 Algebra Review Lesson 5 Radical Expressions 15
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18 ( 3 + 5)( + 4 3)
19 19 ICM Unit 0 Algebra Review SHOW ALL WORK Lesson 5 Homework
20 0
21 1 ICM Unit 0 Algebra Review Lesson 6 Solving Radical Equations There are four steps to solving a radical equation: 1) Isolate the radical. ) Raise both sides to the power of the root. 3) Solve for x. 4) Check for extraneous solution(s). What is an EXTRANEOUS solution? A solution to the final equation but not to the original equation. Extraneous solutions can occur when solving a square root equation but not when solving linear, quadratic or exponential equations. Examples: 1. x = 8. x + 7 = 8 3. x + 6 = x + 11 = 3 5. (x ) 1 3 = x + 5 = 11
22 7. 10x 3 49 = 3x 8. x 6 = 5x x 5 3 = 3x x + 7 = x x = x 1 1. x + = 4 x 13. x + 3 = x x + 8 = x x + 3 = x
23 3 ICM Unit 0 Algebra Review Lesson 6 Homework 1. x 1 = 3. = x 3. 8 x = 0 4. (x + 4) 1 = x 3 = 5 6. x 6 = 3x x = x 8. 9 x 3 = 1 9x 9. 3 x = 1 3x 10. x = (0 x) 1
24 ICM Unit 0 Algebra Review Lesson 7 Solving Rational Equations When solving equations with variables in the denominator, you must check the solution to be sure the denominator will not equal zero. The solution will be eliminated if the denominator is zero. Solve: = 4x+6 5x x+ = = x x 4 x+ 4. 5x x+1 + x = x 9 x x 7 = 5 x 7 6. x x+ = x+3 x+ x x
25 5 ICM Unit 0 Algebra Review SHOW ALL WORK Lesson 7 Homework
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