1. Words To Know Variable Expressions Vocabulary

2. Translating Words to Variable Expressions 1. The SUM of a number and nine2. The DIFFERENCE of a number and nine 3. The PRODUCT of a number and nine4. The QUOTIENT of a number and nine 5. One ninTH OF a number n + 9n – 9 9n9n or 9. A number LESS THAN nine –n9 7. A number SQUARED n2n2 6. Nine times THE QUANTITY OF a number increased by ten. n3n3 8. A number CUBED 9(n + 10) * When you see the phrase less than, reverse the terms.

3. Translating Variable Expressions Translate each mathematical expression into a verbal phrase without using the words:. “plus”, “add”, “minus”, “subtracted”, “”take–away”, multiplied”, “times”, “over”, “power”, or “divided”. 16. 13a 17. 18. y – 11 19. 3y + 8 20. 6 ÷ n 2 21. 7(x + 1) 22. b 3 – 4 the product of thirteen and a number the quotient of fourteen and a number the difference of a number and eleven, OR OR eleven less than a number OR a number decreased by eleven eight more than the product of three and a number, OR the product of three and a number increased by eight the quotient of six and a number squared seven, times the quantity of one more than a number, OR seven, times the quantity of a number increased by one the difference of a number cubed and four, OR a number cubed decreased by four, OR four less than a number cubed

4. Simplifying Using Order of Operations (1 + 3) 2 4 6 – 2 ÷ (–1) Evaluate the numerator and denominator separately ( ) 2 4 6 – 4 4 64 6 – 2 ÷ (–1) (–2) + 6 2 8 16 64 8 = 8 –4 + [ 8 – (5 + 9) ] 2 Evaluate inside the brackets first... –4 + [ 8 – ( 14 ) ] 2 –4 + [ –6 ] 2 –4 + –12 –16...then treat the brackets like parenthesis 1.2.

5. Evaluate each expression using: x = –3 y = –2 z = 6 1) –x –(–3) –(–3) + 3 + 3 3 3 2) –y –(–2) –(–2) + 2 + 2 2 2 3) –z –6 –6 1.Substitute –3 for x only. 2.Leave the negative (–) in front of the x alone. 3.Now, simplify the signs (kill the sleeping man) 1.Substitute –2 for y only. 2.Leave the negative (–) in front of the y alone. 3.Now, simplify the signs. 1.Substitute 6 for z only. 2.Leave the negative (–) in front of the z alone. 3.Now, simplify the signs. Evaluating Variable Expressions with Negative Variables

6. 4) x – y –3 – (–2) –3 – (–2) –3 + 2 –3 + 2 –1 –1 5) x – z –3 – 6 –3 – 6 –9 –9 6) z – x 6 – (–3) 6 – (–3) 6 + 3 6 + 3 9 1.Substitute –3 for x, and –2 for y only. 2.Leave the subtraction sign (–) in front of the y alone. 3.Now, simplify the signs. (keep->change->change) 4.Add the integers. 1. Substitute –3 for x, and 6 for z only. 2. Leave the subtraction sign (–) in front of the z alone. 3. Subtract the integers. 1.Substitute 6 for z, and –3 for x only. 2.Leave the subtraction sign (–) in front of the x alone. 3.Now, simplify the signs. 4.Add the integers. Evaluate each expression using: x = –3 y = –2 z = 6 x = –3 y = –2 z = 6 Evaluating Variable Expressions with Negative Variables

7. 7) xy –3 (–2) –3 (–2) 6 6 8) yz –2 6 –2 6 –12 –12 9) –xz –(–3) 6 –(–3) 6 +3 6 +3 6 18 18 10) –( xz ) –( (–3) 6 ) –( (–3) 6 ) –( –18 ) –( –18 ) 18 18 30) yz –3 – 6 –3 – 6 –9 –9 31) z – x 6 – (–3) 6 – (–3) 6 + 3 6 + 3 9 1.Substitute –3 for x, and –2 for y. 2.Multiply –– * Why? Two variables right next to each other. 1. Substitute –2 for y, and 6 for z. 2. Multiply –– * Why? Two variables right next to each other. 1.Substitute –3 for x, and 6 for z. 2.Leave the negative sign in front of the x alone. 3.Simplify the signs. 4.Multiply 1.Substitute –3 for x, and 6 for z. 2.Leave the negative sign in front of the parenthesis, ( ), alone. 3.Multiply inside the parenthesis first. 4.Simplify the signs. Evaluate each expression using: x = –3 y = –2 z = 6 Evaluating Variable Expressions with Negative Variables

8. 11) 2x 2 2 (–3) 2 2 (–3) 2 2 9 2 9 18 18 12) –2x 2 –2 (–3) 2 –2 (–3) 2 –2 9 –2 9 –18 –18 13) ( –2x ) 2 ( –2 (–3) ) 2 ( –2 (–3) ) 2 ( 6 ) 2 ( 6 ) 2 36 36 1.Substitute –3 for x. 2.First, evaluate the exponent. 3.Then, multiply. –– Why? When a number is right next to a variable, multiply. 1.Substitute –3 for x. 2.First, evaluate the exponent. 3.Then, multiply. 1.Substitute –3 for x. 2.First, evaluate inside parenthesis, ( ). 3.Then, evaluate the exponent. Evaluate each expression using: x = –3 y = –2 z = 6 Evaluating Variable Expressions with Negative Variables

9. 14) 2x 3 2 (–3) 3 2 (–3) 3 2 (–27) 2 (–27) –54 –54 15) –2x 3 –2 (–3) 3 –2 (–3) 3 –2 (–27) –2 (–27) 54 54 16) ( –2x ) 3 ( –2 (–3) ) 3 ( –2 (–3) ) 3 ( 6 ) 3 ( 6 ) 3 216 216 1.Substitute –3 for x. 2.First, evaluate the exponent. (–3) 3 is (–3)(–3)(–3) = –27 * Remember, (–3) 3 is (–3)(–3)(–3) = –27 3. Then, multiply. –– Why? When a number is right next to a variable, multiply. 1.Substitute –3 for x. 2.First, evaluate the exponent. 3.Then, multiply. 1.Substitute –3 for x. 2.First, evaluate inside parenthesis, ( ). 3.Then, evaluate the exponent. Evaluate each expression using: x = –3 y = –2 z = 6 Evaluating Variable Expressions with Negative Variables

10. Evaluating Variable Expressions 7 4 –22 0 11 21 40 59 1. 5.4. 3. 2. 7. 6.

11. Simplifying Variable Expressions by Adding or Subtracting –7a + 11a– 9–3 Circle the variable terms,... – 12 4a... and box up the constants Add the like terms. 1. 17a + a a = 1a Remember, a = 1a so, put a “1” in front of the a 2. –10 –7y + 6y – 3 3. 12b + 5 – 15 – 12b –13–1y... or, get rid of the “1” 0–10... or, get rid of the “0” 4. 14x + 7b – 9x + 19 – 11b – 21 – 4b + 5x – 2 5. 13 + 2(8 – g) Use Distributive Property to get rid of the parenthesis. outer times first, then outer times second 13 + 16 + 2g 29 + 2g 6. 13 +(– 19) – 6(n + 1) – 10n 13 +(– 19) – 6n – 6 – 10n –12 – 16n

12. Simplifying Variable Expressions by Multiplication 8. 7( –3x ) 7 and –3) (x), When you see constants (7 and –3) and variables (x), it’s easiest to simplify them separately. 7( –3x ) 7 3 First, multiply 7 and –3… –21 x …then just bring down the x x (Why? It’s the only x ) x 9. –19a 10bc –19a 10bc When you see constants (–19 and 10) and multiple variables ( a, b, and c ), take it one at a time. –19 10 First, multiply –19 and 10… …then bring down the a, b, and c (Why? There’s only 1 of each.) –190abcabc 10. ( –1 )2y 11. –5a( –5c ) 12. ( x 8 )6y (–1)2y –5a(–5c) (x 8)6y(x 8)6y(x 8)6y(x 8)6y –2y –2y 25ac 25ac 48xy 48xy

13. Simplifying Variable Expressions Using the Distributive Property –2(n +1) DistributiveProperty When a number or variable term sits right next to terms inside parenthesis, use the Distributive Property to simplify. How? –2 n First, multiply the outer term, –2, by the 1 st term in parenthesis, n. –2 n–2 +1 –2n–2n–2–2 Then, multiply the outer term, –2, by the 2 nd term in parenthesis, 1. How to remember the Distributive Property? “Outer times 1 st, then outer times 2 nd ” “Outer times 1 st, then outer times 2 nd ” 14. (9 – 6x)315. –4(8a + 7)16. (–3p + 1)(–5)17. 10(–c – 6) 27 – 18x–32a – 28 15p – 5–10c – 60 13.

14. Are the bases, x, the same? Multiplying Exponents Rule: When multiplying exponent terms with like bases, keep the base, then add the exponents. 4 + 7 = So, we’re going to keep the base... Are we multiplying or dividing the exponent terms? 18. Simplify x 4 x 7 18. Simplify x 4 x 7 x …then add the exponents 11 Rewrite. x 11 19. a 6 a 9 20. b b 5 21. y y 4 y 4 a 15 Careful: What’s the invisible exponent over b ?1 b6b6b6b6 y9y9 Simplifying Variable Expressions by Multiplying Exponents

15. GUIDED PRACTICE Simplifying Variable Expressions by Multiplying Exponents 22. 7x 2 7x 4 7 When you see both constants, 7, and variables, x, it’s easiest to simplify them separately. Simplify 77 x4x4x4x4 x2x2x2x2 7’s Let’s multiply the 7’s first... 49 … then, multiply x 2 x 4. x6x6 23. 10y 7 4y 10y 7 4y 40y 8 24. 3a 5 b 3a 6 b 8 Don’t panic: Just multiply each part separately. 3a5b 3a6b83a5b 3a6b83a5b 3a6b83a5b 3a6b8 9a 11 b 9 25. 2x 5 yz 3 yz 2x 5 yz 3 yz 2x5y2z42x5y2z4

16. Simplifying Variable Expressions by Dividing Exponents Simplify Are the bases, x, the same? Are we multiplying or dividing the exponent terms? Dividing Exponents Rule: When dividing exponent terms with like bases, keep the base, then subtract the exponents. So, we’re going to keep the base... …then subtract the exponents Rewrite. 12 – 7 = x 5 n –6 27. 28. a 4 ÷ a 29.30. (huh?) b a3a3a3a3 y6y6y6y6 or x5x5x5x5 26.

17. Simplifying Variable Expressions by Dividing Exponents 31. Simplify 12 and 6 When you see both constants, 12 and 6, and variables, y, it’s easiest to simplify them separately. Let’s simplify the fraction first... … then, divide y 9 and y 3. 2 y6y6 34. 33. 32. 35.36. 5a5a Hint: The rest of the answers are fractions.