Variable Expressions Vocabulary
Words To Know
Translating Words to 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
n2
9.
8.
A number CUBED
n3
A number LESS THAN nine
n – 9
* When you see the phrase
less than, reverse the terms.
Translating Variable Expressions
Translate each mathematical expression into a verbal phrase without
using the words:
“plus”, “add”, “minus”, “subtracted”, “”take–away”, multiplied”, “times”, “over”, or “divided” .
16.
13a
17.
14
a
the product of thirteen and a number
the quotient of fourteen and a number
eleven less than a number , OR
a number decreased by eleven, OR
the difference of a number and eleven
18.
y – 11
19.
3y + 8 the product of three and a number increased by eight, OR
20.
6 ÷ n2
21.
7(x + 1)
eight more than the product of a number and three
3
22. b – 4
the quotient of six and a number squared
seven, times the quantity of a number increased by one
a number cubed decreased by four
or
four less than a number cubed
or
the difference of a number cubed and four
Simplifying Using Order of Operations
1.
2
(1 + 3) • 4
6 – 2 ÷ (–1)
2
( 4 ) •4
16
•4
64
6 – 2 ÷ (–1)
6 – (–2)
6 + 2
8
64 =
8
8
Evaluate the 2.
Evaluate inside
numerator and –4 + 8 – (5 + 9) •2 the brackets
denominator
first...
–4 + 8 – ( 14 ) •2
separately
[
[
–4 +[
–6
–4 +
]
]
]•2
–12
–16
...then treat
the brackets
like parenthesis
Evaluating Variable Expressions with Negative Variables
1)
–x
Evaluate each expression using:
–(–3)
+3
3
2)
–y
–(–2)
+2
2
3)
x = –3
y = –2
z=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.
–z
–6
–6
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
4)
x–y
–3 – (–2)
–3 + 2
–1
5)
x = –3
1.
2.
3.
4.
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.
x–z
–3 – 6
–9
6)
Evaluate each expression using:
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.
z–x
6 – (–3)
6 +3
9
1.
2.
3.
4.
Substitute 6 for z , and –3 for x only.
Leave the subtraction sign (–) in front of the x alone.
Now, simplify the signs.
Add the integers.
Evaluating Variable Expressions with Negative Variables
xy
7)
–3 • (–2)
6
Evaluate each expression using:
1.
2.
x = –3
y = –2
z=6
Substitute –3 for x , and –2 for y.
Multiply –– * Why? Two variables right next to each other.
yz
8)
–2 • 6
–12
1.
2.
Substitute –2 for y , and 6 for z.
Multiply –– * Why? Two variables right next to each other.
–xz
9)
–(–3) • 6
+3 • 6
18
10)
1.
2.
3.
4.
Substitute –3 for x , and 6 for z.
Leave the negative sign in front of the x alone.
Simplify the signs.
Multiply
1.
2.
3.
4.
Substitute –3 for x , and 6 for z.
Leave the negative sign in front of the parenthesis, ( ), alone.
Multiply inside the parenthesis first.
Simplify the signs.
–(xz)
–((–3) • 6)
–(–18)
18
Evaluating Variable Expressions with Negative Variables
11)
2x2
2 • (–3)2
2• 9
18
12)
x = –3
36
z=6
Substitute –3 for x.
First, evaluate the exponent.
Then, multiply. –– Why? When a number is right next to a
variable, multiply.
1.
2.
3.
Substitute –3 for x.
First, evaluate the exponent.
Then, multiply.
1.
2.
3.
Substitute –3 for x.
First, evaluate inside parenthesis, ( ).
Then, evaluate the exponent.
(–2x)2
(–2 • (–3))2
( 6 )2
y = –2
1.
2.
3.
–2x2
–2 • (–3)2
–2 • 9
–18
13)
Evaluate each expression using:
Evaluating Variable Expressions with Negative Variables
Evaluate each expression using:
14)
1.
2.
3.
–2x3
–2 • (–3)3
–2 • (–27)
54
16)
y = –2
z=6
2 x3
2 • (–3)3
2 • (–27)
–54
15)
x = –3
Substitute –3 for x.
First, evaluate the exponent.
* Remember, (–3)3 is (–3)•(–3)•(–3) = –27
Then, multiply. –– Why? When a number is right next to a
variable, multiply.
1.
2.
3.
Substitute –3 for x.
First, evaluate the exponent.
Then, multiply.
1.
2.
3.
Substitute –3 for x.
First, evaluate inside parenthesis, ( ).
Then, evaluate the exponent.
(–2x)3
(–2 • (–3))3
( 6 )3
216
Evaluating Variable Expressions
1.
2.
4
7
3.
–22
4.
5.
0
11
6.
21
7.
40
59
Simplifying Variable Expressions by Adding or Subtracting
You can only add or subtract
–7a
– 9 + 11a
–3
4a – 12
Circle the variable terms, ...
... and box up the constants
Add the like terms.
1.
4.
17a + a
Remember,
a = 1a
so, put a
“1” in front
of the a
2.
14x + 7b – 9x + 19 – 11b – 21
5.
–10 –7 y + 6 y – 3
–13 –1y
13 + 2(8 – g)
13 + 16 + 2g
– 4b + 5x – 2
29 + 2g
3.
... or, get
rid of the
“1”
Use
Distributive
Property to
get rid of the
parenthesis.
outer times
first, then
outer times
second
12b + 5 – 15 – 12b
0 –10
6.
... or, get
rid of the
“0”
13 +(– 19) – 6(n + 1) – 10n
13 +(– 19) – 6n – 6 – 10n
–12 – 16n
Simplifying Variable Expressions by Multiplication
8.
7( –3x )
7( –3x )
When you see constants (7 and –3) and variables
(x), it’s easiest to simplify them separately.
First, multiply 7 and –3…
…then just bring down the x
(Why? It’s the only x )
–21 x
9.
10.
–19a • 10bc
–19a • 10bc
When you see constants (–19 and 10) and
multiple variables (a, b, and c), take it one at a
time.
–190 abc
First, multiply –19 and 10…
…then bring down the a, b, and c
(Why? There’s only 1 of each.)
( –1 )2y
(–1)2y
–2y
11.
–5a( –5c )
–5a(–5c)
25ac
12.
( x • 8 )6y
(x • 8)6y
48xy
Simplifying Variable Expressions Using the Distributive Property
13.
When a number or variable term sits right next to
terms inside parenthesis, use the
Distributive Property to simplify.
–2(n +1)
–2 •n
–2n
–2 • +1
–2
How?
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.
14. (9 – 6x)3
15. –4(8a + 7)
27 – 18x
–32a – 28
How to remember the Distributive Property?
16. (–3p + 1)(–5)
15p – 5
17. 10(–c – 6)
–10c – 60
“Outer times 1st, then outer times 2nd”
Simplifying Variable Expressions by
18.
Simplify x4 • x7
Multiplying Exponents
Are the bases, x, the same?
Are we multiplying or dividing the exponent
terms?
x
4 + 7 = 11
x
19.
Multiplying Exponents Rule:
When multiplying exponent terms
with like bases, keep the base, then
add the exponents.
a6 • a9
a
15
So, we’re going to keep the base...
…then add the exponents
11
Rewrite.
20.
1
b • b5
b6
Careful:
What’s the
invisible
exponent
over b ?
21.
y • y4 • y4
9
y
Simplifying Variable Expressions by
GUIDED PRACTICE
Multiplying Exponents
Simplify
22.
2
4
7x • 7x
7
x2 •
7
x4
When you see both constants, 7,
and
variables, x, it’s easiest to simplify them
separately.
Let’s multiply the 7’s first...
49 x
23.
6
10y7 • 4y
7
10y • 4y
40y8
… then, multiply x2 • x4 .
24.
3a5b • 3a6b8
Don’t panic:
Just multiply each part
separately.
3a5b • 3a6b8
9a11b9
25. 2x5yz3 • yz
2x5yz3 • yz
2x5y2z4
Simplifying Variable Expressions by Dividing Exponents
Simplify
Are the bases, x, the same?
12
26.
x
7
x
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.
x
12 – 7 = 5
x
5
y9
27.
y3
6
y
So, we’re going to keep the base...
…then subtract the exponents
Rewrite.
28.
4
a ÷a
a
3
9
b
29.
b8
b
n (huh?)
30.
7
n 1
–6
n or n 6
Simplifying Variable Expressions by Dividing Exponents
31. Simplify
When you see both constants, 12 and 6, and
variables, y, it’s easiest to simplify them separately.
12 y 9
6 y3
12 y 9
6 y3
Let’s simplify the fraction
… then, divide y9 and y3 .
2 y6
32.
12first...
6
45a 2 33.
9a
8
32 n
10 n 6
34.
5 p12
15 p 8
35.
28 xy5 z 2 36.
4 xy7
23d 2  d
7f3
Hint: The rest of
the answers are
fractions.
5a
16n
5
2
4
p
3
7z2
y2
23d 3
7f3