1. Exponential Expression
Learning Objective
We will solve equations with squared variables.
What are we going to do?
CFU
Activate Prior Knowledge
Exponential Expression
An exponential expression represents repeated multiplication.
Bases raised to an exponent of 2 are squared.
Evaluate the squared expressions.
1. 42
2. 72 16 49
3. (-4)2
4. (-7)2 16 49
Students, you already know how to evaluate squared expressions. Now, we will use squared expressions solve equations with squared variables.
Make Connection

2. x = 6 and x = -6 both make the equation true.
Concept Development
A square root is a number that is multiplied by itself to form a product.
Taking the square root (√ ) is used to isolate1 the variable in equations with squared variables.
An equation with squared variables can have two solutions, one positive and one negative.
Square Roots
Perfect Squares
12 = 1
22 = 4
32 = 9
42 = 16
52 = 25
62 = 36
72 = 49
82 = 64
92 = 81
102 = 100
x2 = 36
√x2 = √62
x = 6
Which equation below can be solved by taking a square root?
How do you know?
A y2 = 8
B y2 = 4
Explain why 3 are the solutions to the equation z2 = 9
CFU
Solutions
(6)2 = 36
(-6)2 = 36
x = 6 and x = -6 both make the equation true.
1 separate (synonym)
Vocabulary

3. Skill Development/Guided Practice
A square root is a number that is multiplied by itself to form a product.
Taking the square root (√ ) is used to isolate the variable in equations with squared variables.
An equation with squared variables can have two solutions, one positive and one negative.
Rewrite the numerical term as a squared expression.
Solve the equation by taking the square root of both sides of the equation.
Check and interpret2 the solution(s).
Solve equations with squared variables. 1 2 3
How did I/you solve the equation?
How did I/you check and interpret the solution(s)?
CFU 2 3
1. a2 = 81
2. b2 = 25
3. h2 = 36
4. k2 = 16
5. q2 =
6. p2 =
7. w2 =
8. m2 =
92 = 81
52 = 25
a2 = 92
(-9)2 = 81
b2 = 52
(-5)2 = 25
√a2 = √92
√b2 = √52
Perfect Squares
12 = 1
22 = 4
32 = 9
42 = 16
52 = 25
62 = 36
72 = 49
82 = 64
92 = 81
102 = 100
a = 9 and a = -9 both make the equation true.
b = 5 and b = -5 both make the equation true.
a = 9
b = 5
62 = 36
42 = 16
h2 = 62
(-6)2 = 36
k2 = 42
(-4)2 = 16
√h2 = √62
√k2 = √42
h = 6 and h = -6 both make the equation true.
k = 4 and k = -4 both make the equation true.
h = 6
k = 4 9
100
32 = 9
102 = 100
q2 = 32
(-3)2 = 9
p2 = 102
(-10)2 = 100
√q2 = √32
√p2 = √102
q = 3 and q = -3 both make the equation true.
p = 10 and p = -10 both make the equation true.
q = 3
p = 10 64 49
82 = 64
72 = 49
w2 = 82
(-8)2 = 64
m2 = 72
(-7)2 = 49
√w2 = √82
√m2 = √72
w = 8 and w = -8 both make the equation true.
m = 7 and m = -7 both make the equation true.
2 explain (synonym)
Vocabulary
w = 8
m = 7

4. 8 in s2 = 64 82 = 64 s2 = 82 (-8)2 = 64 8 in Area = 64 in2 √s2 = √82
Skill Development/Guided Practice (continued)
How did I/you determine what the question is asking?
How did I/you determine the math concept required?
How did I/you determine the relevant information?
How did I/you solve and interpret the problem?
How did I/you check the reasonableness of the answer?
CFU 2 1 3 4 5
8 in
9. Cameron is tiling his floor with square tiles. He knows the area of each tile is 64 in2. What is the length of each side of the tile?
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10. Vanessa is starting a painting on a square canvas. The area of the canvas is 81 in2. What is the length of each side of the canvas?
s2 = 64
82 = 64
s2 = 82
(-8)2 = 64
8 in
Area = 64 in2
√s2 = √82
s = 8
The length of each side of the tile is 8 in. (-8 in is not a solution because a length cannot be negative.)
9 in
s2 = 81
92 = 81
s2 = 92
(-9)2 = 81
9 in
Area = 81 in2
√s2 = √92
s = 9
The length of each side of the canvas is 9 in. (-9 in is not a solution because a length cannot be negative.)

5. Relevance
A square root is a number that is multiplied by itself to form a product.
Taking the square root (√ ) is used to isolate the variable in equations with squared variables.
An equation with squared variables can have two solutions, one positive and one negative. 1
Solving equations with squared variables will help you solve real-world problems.
How long is the Salinas River from King City to Soledad?
King City
Salinas River
Soledad
16 mi
12 mi
The Salinas River is 20 miles long from King City to Soledad.
= r2
= r2
√400 = √r2
20 = r 2
Solving equations with squared variables will help you do well on tests.
Sample Test Question:
87. Choose Yes or No to indicate whether each statement is true about the equation w2 = 4.
A The equation can be solved by taking a square root.
B w = 16
C w = 2
D There are exactly two solutions to the equation.
O Yes O No
Does anyone else have another reason why it is relevant to solve equations with squared variables? (Pair-Share) Why is it relevant to solve equations with squared variables? You may give one of my reasons or one of your own. Which reason is more relevant to you? Why?
CFU

6. A square root is a number that is multiplied by itself to form a product.
Taking the square root (√ ) is used to isolate the variable in equations with squared variables.
An equation with squared variables can have two solutions, one positive and one negative.
Word Bank
Skill Closure
Rewrite the numerical term as a squared expression.
Solve the equation by taking the square root of both sides of the equation.
Check and interpret the solution(s).
Solve equations with squared variables. 1 2 3
Perfect Squares
12 = 1
22 = 4
32 = 9
42 = 16
52 = 25
62 = 36
72 = 49
82 = 64
92 = 81
102 = 100
1. d2 = 49
2. n2 =
72 = 49 81
92 = 81
d2 = 72
(-7)2 = 49
n2 = 92
(-9)2 = 81
√d2 = √72
√n2 = √92
d = 7 and d = -7 both make the equation true.
n = 9 and n = -9 both make the equation true.
d = 7
n = 9
Access Common Core
Word Bank
equation
square root
solution
Which equation below can be solved by taking a square root? Explain your answer. Explain why the other equations cannot be solved by taking a square root.
A x2 = 25 B y2 = 16 C z2 = 13
Summary Closure
What did you learn today about solving equations with squared variables? (Pair-Share)
Use words from the word bank.

7. Independent Practice
A square root is a number that is multiplied by itself to form a product.
Taking the square root (√ ) is used to isolate the variable in equations with squared variables.
An equation with squared variables can have two solutions, one positive and one negative.
Rewrite the numerical term as a squared expression.
Solve the equation by taking the square root of both sides of the equation.
Check and interpret the solution(s).
Solve equations with squared variables. 1 2 3
1. g2 = 36
2. u2 = 16
3. c2 = 1
4. v2 = 49
62 = 36
42 = 16
g2 = 62
Perfect Squares
12 = 1
22 = 4
32 = 9
42 = 16
52 = 25
62 = 36
72 = 49
82 = 64
92 = 81
102 = 100
(-6)2 = 36
u2 = 42
(-4)2 = 16
√g2 = √62
√u2 = √42
g = 6
u = 4
g = 6 and g = -6 both make the equation true.
u = 4 and u = -4 both make the equation true.
12 = 1
72 = 49
c2 = 12
(-1)2 = 1
v2 = 72
(-7)2 = 49
√c2 = √12
√v2 = √72
c = 1
v = 7
c = 1 and c = -1 both make the equation true.
v = 4 and v = -4 both make the equation true.

8. 5 ft 5 ft s2 = 25 52 = 25 s2 = 52 (-5)2 = 25 √s2 = √52 s = 5
Independent Practice (continued)
5 ft
5. A square portion of Carmen’s apartment is tiled. Each tile is 1 ft2, and there are 25 tiles. How long is each side of the tiled portion of Carmen’s apartment?
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5 ft
s2 = 25
52 = 25
s2 = 52
(-5)2 = 25
√s2 = √52
s = 5
The length of each side of the tiled portion of Carmen’s apartment is 5 ft.
(-5 ft is not a solution because a length cannot be negative.)

9. Periodic Review 1
Extension:
Describe in your own words what it means to take the square root of a number:
_______________________________________________________________________________________________________________________
How is squaring a number different from taking the square root of a number?

10. Find the square of each number below:
Periodic Review 1
Extension:
Find the square of each number below: