Name: __________________________
Teacher: ________________________
Order of Operations
The order of operations is a set of guidelines that make it possible to be sure that two people will get the same result
when evaluating an expression. Without this standard order of operations, two people might evaluate an expression
differently and arrive at different values. For example, without the order of operations, someone might evaluate all
expressions from left to right, while another person performs all additions and subtractions before all multiplications
and divisions.
You can use the acronym P.E.M.A. (Parentheses, Exponents, Multiplication and Division, and Addition and
Subtraction) to help you remember the order of operations.
How do you evaluate the expression 3 + 4 × 2 - 10 ÷ 5?
3 + 8 – 10 ÷ 5
= 3+ 8 – 2
= 11 – 2
=9
There are no parentheses or exponents, so first, do
any multiplication or division from left to right.
Do any addition or subtraction from left to right.
Exercises
Simplify each expression.
1. 8 + 7 · 9
2. 12 + 42
3. 35 – (17 – 2) ÷ 5
4. 24 – 9 · 2 + 6 ÷ 3
7. 26 – [ (25 – 11) – 23]
8. (82 – 25) ÷ (24 ÷ 6) + 32
9. (5 + 3)2
11. (15 – 3) ÷ 4
10. (8 – 5) (14 – 6)
 22  3 
 5 
12. 
13. 40 – 15 ÷ 3
14. 20 + 12 ÷ 2 – 5
15. (42 + 52)2
16. 4 ×5 – 32 × 2 ÷ 6
Write and simplify an expression to model the relationship expressed in the situation below.
17. Manuela has two boxes. The larger of the two boxes has dimensions of 15 cm by 25 cm by 20 cm. The smaller of the two
boxes is a cube with sides that are 10 cm long. If she were to put the smaller box inside the larger, what would be the
remaining volume of the larger box?
Equations of Lines in the Coordinate Plane
To find the slope m of a line, divide the change in the y values by the change in the x values from one point to another.
rise change in y
Slope is
or
.
run change in x
What is the slope of the line through the points (5, 3) and (4, 9)?
Slope 
change in y
93
6 2

= =
change in x
4   5 9 3
Exercises
Find the slope of the line passing through the given points.
1. (5, 2), (1, 8)
2. (1, 8), (2, 4)
3. (2, 3), (2, 4)
If you know two points on a line, or if you know one point and the slope of a line, then you can find the
equation of the line.
Write an equation of the line that contains the points J(4, 5) and
K(2, 1). Graph the line.
If you know two points on a line, first find the slope using
m 
m 
y2  y1
.
x2  x1
1   5 
2  4

6
 1
6
Now you know two points and the slope of the line. Select one
of the points to substitute for (x1, y1). Then write the equation
using the point-slope form y  y1 = m(x  x1).
y  1 = 1(x  (2))
Substitute.
y  1 = 1(x + 2)
Simplify within parentheses. You may leave your equation in this form or further
simplify to find the slope-intercept form.
y  1 = x  2
y = x  1
Answer: Either y  1 = 1(x + 2) or y = x  1 is acceptable.
Exercises
Graph each line.
4. y 
1
x4
2
1
3
5. y  4  ( x  3)
6. y  3 = 6(x  3)
If you know two points on a line, or if you know one point and the slope of a line, then you can find the
equation of the line using the formula y  y1 = m(x  x1).
Use the given information to write an equation for each line.
7. slope 1, y-intercept 6
8. slope
4
, y-intercept 3
5
9.
10.
11. passes through (7, 4) and (2, 2)
12. passes through (3, 5) and (6, 1)
Graph each line.
13. y = 4
14. x = 24
15. y = 22
Write each equation in slope-intercept form.
16. y  7 = 2(x  1)
1
3
3
2
18. y  5   ( x  3)
17. y  2  ( x  5)
Find the solution to the system.
19.
20.
Multiplying Powers With the Same
When multiplying powers with the same base, you add the exponents. This
is true for numerical and algebraic expressions.
Problem
What is each expression written as a single power? a. 34 · 32 · 33
All three powers have the same base, so this expression can be written as a single power by adding the
exponents.
All powers have the same base. Add the exponents.
34 · 32 · 33 = 34+2+3
=39
Simplify the exponent.
34 represents 4 factors of 3, 32 represents 2 factors of 3, and 33 represents 3 factors of 3. This is a total of 9
factors of 3, so the answer is reasonable.
Even when some of the exponents are negative, exponents can be added when the bases are the same in a
product of powers.
b. 113 · 114 · 115
113 · 114 · 115 = 113 + 4 + (5)
All powers have the same base. Add the exponents.
= 114
Simplify the exponent.
Exercises
Simplify each expression.
1. a2a3
2. 3n3n5
3. 8k3 · 3k6
4. (8p5)(6p4)
5. 21d7 · 2d3
6. (6.1m4)(3m2)
7. h5 ·h2 ·h10
8. (9q8)(6q11)
9. (16r7)( 2r)
10. (y3z13)(y2z6)
13. m6 ·m3 ·n2
11. (3x2)(5w8)(4x3)
14. 6j3k · 7jk5
12. (15fg2)(f 3g3)( 8f1g6)
15. 2uvw1 · 3u2v2w
When a power is raised to another power, like (xy)z, multiply the exponents.
Problem
What is the simplified form of (d3)4?
(d3)4 = d3·4
= d12
The expression is a power, d3, raised to another power, 4. Multiply the
exponents.
Simplify.
Simplifying powers may require you to use multiple properties of exponents. You should follow the
order of operations when simplifying exponential expressions.
Problem
What is the simplified form of (n–3)6n4?
Using the order of operations, first simplify the power (n–3)6.
(n–3)6n4 = (n–3·6)n4 = n–18n4
Next, multiply. The two powers have the same base, so simplify by adding the exponents.
n–18n4 = n–18+4 = n–14
Finally, write the expression using positive exponents. Rewrite the expression using the reciprocal of the base and
the opposite of the exponent.
n 14 
1
n14
Exercises
Simplify each expression.
1. (y2)3
2. (v9)6
3. (h4)5
4. (n4)11
5. (p–1)5
6. (z3)–6
7. (x–4)–5x
8. (f 5)–1f 8
9. (3a)4
10. (6c)–3
11. (7k)0
12. (10s–3)2
13. (2y–5)3(x11y–10)2
14. u–9(u–1v)4u–5
15. (x13y6)–2(y–5x10)6
16. 4m0n0(6m5)2
Division Properties of Exponents
What is 66 divided by 64 ?
Method 1: Evaluate 66 and 64, and then divide the results.
66 = 46,656
64 = 1296
46,656 ÷ 1296 = 36
Method 2: Expand the numerator and denominator.
66 6 6 6 6 6 6

64
6666
After dividing out the common factors, you are left with 6 × 6 = 36.
When you divide powers with the same base, subtract the exponents. In the example above, 66 and 64 are
powers with the same base and when you divided them, the result was 36 = 62. This is the same result
you get by subtracting the
exponents:
66
 66  4  6 2 .
4
6
The division property of exponents also allows you to simplify quotients that contain variables.
How can you use the division property of exponents to show that x 2 
x5
when x ≠ 0?
x3
Expand the numerator and denominator.
x5 x  x  x  x  x

x3
x x  x
After dividing out the common factors you are left with x · x = x2.
Division properties of exponents work whether the bases in the problem are constants or variables. When you divide
powers with the same base, subtract the exponents. In this example, x5 and x3 are powers with the same base and when
you divided them, the result was x2 = x5–3.
Simplify each expression.
75
72
2.
52
5
m4
5. 2
m
r3
7.
r
a3
9. 5
a
4z
44
p6
6. 5
p
5 4
x y
8. 3
x y
10 x 5
10.
15 x 2
1.
3.
4.
39
32
Homework: Adding and Subtracting Integers
You can do it.
This part of your brain tells you how.
Use a number line or the “rules” to simplify each expression below.
For like sign, add the absolute values and keep the sign.
For different signs, subtract the absolute values and keep the sign of the larger absolute value.
1. 8 + (-5) =
11. -5 + (-4) =
21. 14 + 7 + (-4) =
2. -6 – 8 =
12. 13 + (-25)
3. 15 + (-3) =
13. -4 + 4 =
23. -6 + 10 + (-8) =
4. 7 + (- 8) =
14. -10 + (-10) =
24. 23 + (-6) + 2 =
5. 14 – 17 =
15. 9x + (-4x) + 3x =
25. 4 + (-7) + (-8)=
6. 3x + 12x =
16. -11x + (-9x) =
26. -12x + (-4x) + (-5x) =
7. (-3) + (-3) =
17. 5x + 8x + (-5x) =
27. -8 – 7 – 12 =
8. 11 + (-14) =
18. 4a + 9a + (-13a) =
28. 5 – 12 + 7 =
9. -12 + (-18) =
19. 20x + (- 9x) + 3x =
29. 6 + (-4) + (-9) + 7 =
10. 1 + (-5) =
20. 6x + 12x + (-7x) =
30. -9 + (-5) + 14 + (-6) =
=
22. -9 + (- 4) + (-3) =
Evaluate each expression.
Remember that - (-) = +. There must be no number between the two negatives.
31. 14 – (-5) =
33. –15 – (-7) =
32. –(-7) + 18 =
34. 12x – (-4x) =
35. 3x + 9x – (-4x) =
36. -8x – (-3x) –(-2x) =
Substitute the given value for the variable and evaluate. x = - 4, y = -7, m = 3
37. x + y + 5 =
38. m – ( x) =
39. – (-y) + x =