Variable Expressions Vocabulary Words To Know

p.3 – Evaluating Variable Expressions Translating Words to Variable Expressions p.3 – Evaluating Variable Expressions p.3 – Evaluating Variable Expressions 1. The SUM of a number and nine 2. The DIFFERENCE of a number and nine n + 9 n – 9 3. The PRODUCT of a number and nine 4. The QUOTIENT of a number and nine 9n 5. One ninTH OF a number 6. Nine times THE QUANTITY OF a number increased by ten. 9(n + 10) or 7. A number SQUARED 8. A number CUBED n2 n3 * When you see the phrase less than, reverse the terms. 9. A number LESS THAN nine n – 9 Writing Variable Expressions, p.1

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 ÷ n2 21. 7(x + 1) 22. b3 – 4 the product of thirteen and a number the quotient of fourteen and a number the difference of a number and eleven, 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

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

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

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

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

Evaluating Variable Expressions with Negative Variables 2 • (–3)2 2 • 9 18 12) –2x2 –2 • (–3)2 –2 • 9 –18 13) (–2x)2 (–2 • (–3))2 ( 6 )2 36 Evaluate each expression using: x = –3 y = –2 z = 6 Substitute –3 for x. First, evaluate the exponent. Then, multiply. –– Why? When a number is right next to a variable, multiply. Then, multiply. First, evaluate inside parenthesis, ( ). Then, evaluate the exponent.

Evaluating Variable Expressions with Negative Variables Evaluate each expression using: x = –3 y = –2 z = 6 14) 2x3 2 • (–3)3 2 • (–27) –54 15) –2x3 –2 • (–3)3 –2 • (–27) 54 16) (–2x)3 (–2 • (–3))3 ( 6 )3 216 Substitute –3 for x. First, evaluate the exponent. * Remember, (–3)3 is (–3)•(–3)•(–3) = –27 3. Then, multiply. –– Why? When a number is right next to a variable, multiply. Then, multiply. First, evaluate inside parenthesis, ( ). Then, evaluate the exponent.

Evaluating Variable Expressions 1. 2. 4 7 3. –22 4. 5. 11 6. 40 21 59 7.

18a –7a + 11a – 9 –3 4a – 12 1 –13 –1y –10 – 4b + 5x – 2 29 + 2g Simplifying Variable Expressions by Adding or Subtracting You can only add or subtract LIKE TERMS. –7a + 11a – 9 –3 terms with the same variable. numbers without variables (constants) or (like –7a and +11a) (like –9 and –3) Circle the variable terms, ... 4a – 12 ... and box up the constants Add the like terms. 1 1. 17a + a Remember, a = 1a so, put a “1” in front of the a 2. –10 –7y + 6y – 3 3. 12b + 5 – 15 – 12b ... or, get rid of the “1” ... or, get rid of the “0” 18a –13 –1y –10 Use Distributive Property to get rid of the parenthesis. 4. 14x + 7b – 9x + 19 – 11b – 21 5. 13 + 2(8 – g) 6. 13 +(– 19) – 6(n + 1) – 10n 13 +(– 19) – 6n – 6 – 10n 13 + 16 + 2g outer times first, then outer times second – 4b + 5x – 2 29 + 2g –12 – 16n

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

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

x x11 y9 a15 Are the bases, x, the same? Simplifying Variable Expressions by Multiplying Exponents 18. Simplify x4 • x7 Are the bases, x, the same? Are we multiplying or dividing the exponent terms? Multiplying Exponents Rule: When multiplying exponent terms with like bases, keep the base, then add the exponents. 4 + 7 = 11 So, we’re going to keep the base... x …then add the exponents x11 Rewrite. 1 Careful: What’s the invisible exponent over b ? 19. a6 • a9 20. b • b5 21. y • y4 • y4 y9 b6 a15

Simplifying Variable Expressions by Multiplying Exponents GUIDED PRACTICE Simplify 22. 7x2 • 7x4 When you see both constants, 7, and variables, x, it’s easiest to simplify them separately. 7 7 x2 • x4 Let’s multiply the 7’s first... 49 x6 … then, multiply x2 • x4 . 23. 10y7 • 4y 24. 3a5b • 3a6b8 25. 2x5yz3 • yz Don’t panic: Just multiply each part separately. 10y7 • 4y 2x5yz3 • yz 40y8 2x5y2z4 3a5b • 3a6b8 9a11b9

Simplify x x5 28. a4 ÷ a 27. 29. 30. y6 a3 b n–6 or Simplifying Variable Expressions by Dividing Exponents Simplify Are the bases, x, the same? 26. 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. 12 – 7 = 5 So, we’re going to keep the base... x …then subtract the exponents x5 Rewrite. 28. a4 ÷ a 27. 29. 30. (huh?) y6 a3 b n–6 or

Simplifying Variable Expressions by Dividing Exponents 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 y9 and y3 . 2 y6 32. 33. 35. 36. 34. Hint: The rest of the answers are fractions. 5a