1-4 Simplifying Algebraic Expressions To evaluate an algebraic expression you substitute numbers for variables. Then follow the order of operations. Here is a sentence that can help you remember the order of operations. Please Excuse My Dear Aunt Sally Parentheses Exponents Multiply Divide Add Subtract Evaluate x 2xy y 2 for x 4 and y 6. 4 2 4 6 6 2 Substitute 4 for x and 6 for y. 4 2 4 6 36 Evaluate exponents: 6 2 36. 4 48 36 Multiply from left to right. 8 Add and subtract from left to right. Evaluate each expression for the given values of the variables. 1. a 2 2a b 2 3a for a 5 and b 2 5 2 2 5 2 2 3 5 Substitute 5 for a and 2 for b. 25 + 2 5 4 3 5 Evaluate exponents. 25 40 15 Multiply from left to right. 50 Add and subtract from left to right. 2. c 2 cd 3d for c 7 and d 6 7 2 7 6 3 6 Substitute for the variables. 49 7 6 3 6 Evaluate exponents. 49 42 18 Multiply from left to right. 25 Add and subtract from left to right. 3. 5m n 3 for m 4 and n 2 3n 5 4 2 3 3 2 Substitute for the variables. 5 4 8 3 2 20 8 3 2 12 3 2 12 6 Evaluate exponents inside parentheses. Multiply inside parentheses. Subtract inside parentheses. Multiply from left to right. 2 Divide from left to right. 30 Holt Algebra 2
1-4 Add or subtract the coefficients of like terms to simplify an algebraic expression. Like Terms: 3 x 2 and 4 x 2 5xy and xy 3 x 2 5xy 4 x 2 xy 2 3 x 2 4 x 2 5xy xy 2 Group like terms. 7 x 2 4xy 2 Add or subtract like terms. You can use the Distributive Property to simplify an algebraic expression. 2( a 2 ab) 6ab 2a 2 2 a 2 2ab 6ab 2a 2 Distribute. 2(a 2 ab) 2( a 2 ) 2( ab) 2 a 2 2ab 2 a 2 2a 2 2ab 6ab 8ab Simplifying Algebraic Expressions (continued) Simplify each expression. 4. 6x 3 2x 4x 6x 2x 4x 3 4x 3 5. c (4c d ) c 2 cd Group like terms. Coefficients of x 2 : 3 and 4 Coefficients of xy : 5 and 1 Add or subtract like terms. Group like terms. Add or subtract like terms. 4c 2 cd c 2 cd Distribute. 4c 2 c 2 cd cd Group like terms. 3 c 2 2cd Add or subtract like terms. 6. 4a 2 5ab 4a 2 2ab 7 7. 3 s 4t 3s t Think: 3x 2 4x 2 7 x 2 5xy 1xy 4xy Think: 2a 2 2a 2 0 3ab 7 6s 13t 31 Holt Algebra 2
1-5 Properties of Exponents Write Expanded Form Exponent Form Read a a a 2 a squared a a a a 3 a cubed a a a a a 4 a to the fourth power a a 4 a a n a to the nth power 4 x 5 4 x x x x x 4 x 5 4x 4x 4x 4x 4x 4x 5 4x 4x 4x 4x 4x 4 x 3 y 6 2 4 x x x y 6 y 6 Zero Exponent Property: a 0 1; a is not zero 38 0 1 Negative Exponent Property: a n 1 and a a n b n b a n ; a is not zero. 3 4 1 3 1 4 3 3 3 3 1 81 2 5 3 5 2 3 5 2 5 2 5 2 125 8 Write each expression in expanded form. List the factors to expand exponential expressions. 1. 8 c 3 2. 3xy 4 3. a 3 b c 2 8 c c c 3xy 3xy 3xy 3xy a a a b c b c Evaluate each expression. 4. 6 1 5. 10 0 6. 12 2 1 6 1 1 144 7. 4 3 8. 1 7 2 9. 3 4 3 1 64 49 64 27 10. 5 0 11. 2 5 2 12. 1 3 2 1 4 25 9 38 Holt Algebra 2
1-5 Properties of Exponents (continued) Properties of Exponents (m and n are integers; a and b are nonzero real numbers.) Same Base: a m a n a m n a m a m n a n a m n a m n To multiply, add exponents. To divide, subtract exponents. To raise to a power, multiply exponents. Different Bases: ab m a m b m a b m a m b m Combine properties of exponents to simplify expressions with exponents. 2 x 5 4 Distribute the exponent. c 4 d c 3 d 2 2 4 x 5 4 Distribute the exponent. c 4 c 3 d d 2 Group like variables. 2 4 x 5 4 Multiply exponents. c 4 3 d 1 2 Add exponents. 2 4 x 20 Simplify. cd 3 Simplify. 16 x 20 3r s 5 r 4 s 3 3 r 1 4 s 5 3 Subtract exponents. 3 r 3 s 2 Simplify. 3 s 2 r 3 Simplify each expression. Assume all variables are nonzero. Record answer with positive exponents. 13. 5a b 3 2 14. w 2 x 5 w 4 x 3 15. y 4 z 3 y 1 z 2 5 2 a 2 b 3 2 w 2 4 x 5 3 y 4 y 2 z 3 z 2 25 a 2 b 6 w 6 x 8 y 2 z 5 16. 6 s 3 t 3s t 17. a 4 2 b 2 6 3 s 3 1 4 3 1 2 t a 2 s 2 t b 2 3 3 a 12 18. 3 x 2 y 3 9 x 1 y 5 3 9 x 2 1 y 3 5 27 y 8 b 6 x 3 39 Holt Algebra 2
2-1 Solving Linear Equations and Inequalities Use the Distributive Property to solve equations. 8 y 6 64 8y 48 64 Combine like terms to solve equations. 48 48 8y 112 8y 8 112 8 y 14 4x 18 3 3x 45 5x 4x 15 8x 45 4x 4x 15 4x 45 45 45 60 4x 60 4 4x 4 15 x Distribute the 8 to both terms. Think: Add 48 to both sides. Divide both sides by 8. 3x and 5x are like terms. Subtract 4x from both sides. Add 45 to both sides. Divide both sides by 4. Solve. 1. 3 x 9 63 2. 7 y 4 98 3. 8 w 6 168 3x 63 7y 98 168 3x 7y x y w 4. 5a 3 2a 9 5. 8y y 3y 30 6. x 5 29 3x 2a 2a 3y 30 3x 3x 3a 3 29 3a 30 a y x 6 Holt Algebra 2
2-1 Solving Linear Equations and Inequalities (continued) Reverse the inequality symbol if you multiply or divide both sides by a negative number. Combine like terms. Graph the solution: x 3 x 3 5x 9 5x 5x 4x 3 9 3 3 4x 12 4x 4 12 4 x 3 x and 5x are like terms. 3 and 9 are like terms. Divide by 4. Reverse the inequality symbol. The symbol means that 3 is included in the graph. Substitute test values into the original inequality to check: Pick a value that should be a solution. Try x 0. 0 3 5 0 9? 3 9 True Pick a value that should NOT be a solution. Try x 4. 4 3 5 4 9? 7 11 False Solve each inequality. Check your solutions. 7. 2x 8x 24 8. 5w 2 8w 23 8x 8x 8w 8w 23 9. 7y 1 12y 29 29 24 x 24 w y x w y 7 Holt Algebra 2