1. Variable Expressions Vocabulary
Words To Know

2. 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

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” . a
17.
y – 11
y + 8
÷ n2
(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

4. –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 +[ – ]•2
...then treat the brackets like parenthesis
( )2• 4 4
• 4 16
– –12 64
6 – 2 ÷ (–1)
–16
6 –
(–2) + 8 64 8 = 8

5. Evaluating Variable Expressions with Negative Variables
–(–3)
+ 3 3
2) –y
–(–2)
+ 2 2
3) –z –6
Evaluate each expression using:
x = – y = – 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.

6. 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 = – y = – 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.

7. 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 = – y = – z = 6
Substitute –3 for x , and –2 for y.
Multiply –– * Why? Two variables right next to each other.
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.

8. 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 = – y = – 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.

9. Evaluating Variable Expressions with Negative Variables
Evaluate each expression using: x = – y = – z = 6
14) x3
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.

10. Evaluating Variable Expressions1. 2. 4 7 3.
–22 4. 5. 11 6. 40 21 59 7.

11. 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 a
– 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
a + a
Remember, a = 1a
so, put a “1” in front of the a
2. –10 –7y + 6y – 3
b + 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.
x + 7b – 9x + 19 – 11b – 21
(8 – g)
(– 19) – 6(n + 1) – 10n
13 +(– 19) – 6n – 6 – 10n g
outer times first, then outer times second
– 4b + 5x – 2
29 + 2g
–12 – 16n

12. 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.
( –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.)
( –1 )2y
–5a( –5c )
( x • 8 )6y
–5a(–5c)
(x • 8)6y
(–1)2y
25ac
48xy
–2y

13. 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)
(–c – 6)
27 – 18x
–32a – 28
15p – 5
–10c – 60
How to remember the Distributive Property?
“Outer times 1st, then outer times 2nd”

14. x x11 y9 a15 Are the bases, x, the same?
Simplifying Variable Expressions by Multiplying Exponents
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 ?
a6 • a9
b • b5
y • y4 • y4 y9 b6
a15

15. Simplifying Variable Expressions by Multiplying Exponents
GUIDED PRACTICE
Simplify
x2 • 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 .
y7 • 4y
24. 3a5b • 3a6b8
x5yz3 • yz
Don’t panic:
Just multiply each part separately.
10y7 • 4y
2x5yz3 • yz
40y8
2x5y2z4
3a5b • 3a6b8
9a11b9

16. 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.
a4 ÷ a
27.
29.
30.
(huh?) y6 a3 b
n–6 or

17. 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