1. 1. √49 2. –√144 Lesson 4.5, For use with pages 266-271
Find the exact value.
1. √49
ANSWER 7
2. –√144
ANSWER
–12

2. 3. Use calculator to approximate the value of to the nearest tenth.
Lesson 4.5, For use with pages 16
3. Use calculator to approximate the value of
to the nearest tenth. 82
ANSWER
2.3
4. The area of half of a square mural is square feet. What is the length of a side of the mural? 1 2
ANSWER
11 ft

3. 4.5 Properties of Square Roots

4. EXAMPLE 1
Use properties of square roots
Simplify the expression. a. 80 5 16 = 5
= 4 b. 6 21
126 = 9 14 =
= 3 14 c. 4 81 = 4 81 =
2 9 d. 7 16 = 7 16 = 4 7

5. GUIDED PRACTICE
GUIDED PRACTICE
for Example 1 27
SOLUTION 27 3 9 = 3
= 3 98
SOLUTION 98 2 49 = 2
= 7

6. GUIDED PRACTICE
GUIDED PRACTICE
for Example 1 10 15
SOLUTION 10 15 6 25
150 = 6
= 5 8 28
SOLUTION 8 28
224 = 14 16 = 14
= 4

7. GUIDED PRACTICE
GUIDED PRACTICE
for Example 1 9 64
SOLUTION 9 64 9 64 = =
3 8 15 4
SOLUTION 15 4 15 4 = = 15 2

8. GUIDED PRACTICE
GUIDED PRACTICE
for Example 1 11 25
SOLUTION 11 25 11 25 = = 5 11 36 49
SOLUTION 36 49 36 49 = 7 6 =

9. EXAMPLE 2
Rationalize denominators of fractions. 5 2 3
7 + 2
Simplify
(a)
and
(b)
SOLUTION
(a) 5 2 = 5 2 = 5 2 2 10 =

10. EXAMPLE 2
Rationalize denominators of fractions.
SOLUTION = 3
7 + 2
7 –
(b) 3
7 + 2 =
21 – 3 2
49 – – 2 =
21 – 3 2 47

11. Write original equation.
EXAMPLE 3
Solve a quadratic equation
Solve 3x2 + 5 = 41.
3x2 + 5 = 41
Write original equation.
3x2 = 36
Subtract 5 from each side.
x2 = 12
Divide each side by 3.
x = + 12
Take square roots of each side.
x = + 4 3
Product property
x =
+ 2 3
Simplify.

12. EXAMPLE 3
Solve a quadratic equation
ANSWER
The solutions are and 2 3 2 3 –
Check the solutions by substituting them into the original equation.
3x2 + 5 = 41
3x2 + 5 = 41
3( )2 + 5 = 41 ? 2 3
3( )2 + 5 = ?
– 2 3
3(12) + 5 = 41 ?
3(12) + 5 = 41 ?
41 = 41 
41 = 41 

13. Write original equation.
EXAMPLE 4
Standardized Test Practice
SOLUTION 15
(z + 3)2 = 7
Write original equation.
(z + 3)2 = 35
Multiply each side by 5.
z + 3 = + 35
Take square roots of each side.
z = – 3 + 35
Subtract 3 from each side.
The solutions are – and – 3 – 35

14. EXAMPLE 4
Standardized Test Practice
ANSWER
The correct answer is C.

15. GUIDED PRACTICE
GUIDED PRACTICE
for Examples 2, 3, and 4
Simplify the expression. 6 5
SOLUTION 6 5 = 6 5 = 6 5 5 30 =

16. GUIDED PRACTICE
GUIDED PRACTICE
for Examples 2, 3, and 4
Simplify the expression. 9 8
SOLUTION 9 8 = 9 8 = 9 8 2 4 = 3

17. GUIDED PRACTICE
GUIDED PRACTICE
for Examples 2, 3, and 4
Simplify the expression. 17 12
SOLUTION 17 12 = 17 12 = 17 12 51 12 = 2 6

18. GUIDED PRACTICE
GUIDED PRACTICE
for Examples 2, 3, and 4
Simplify the expression. 19 21
SOLUTION 19 21 = 19 21 = 19 21
399 21 =

19. GUIDED PRACTICE
for Examples 2, 3, and 4
– 6
7 – 5
SOLUTION =
– 6
7 – 5
7 +
– 6
7 – 5 =
– 42 – 6 5
– – 5 =
– 21 – 3 5 22

20. GUIDED PRACTICE
for Examples 2, 3, and 4 2
4 + 11
SOLUTION = 2
4 + 11
4 – 2
4 + 11 =
8 – 2 11
– – 11 =
8 – 2 11 5

21. GUIDED PRACTICE
for Examples 2, 3, and 4
– 1
9 + 7
SOLUTION
– 1
9 + 7 =
9 – =
– 9 + 7
– – 7 =
– 9 + 7 74

22. GUIDED PRACTICE
for Examples 2, 3, and 4 4
8 – 3
SOLUTION = 4
8 – 3
8 + 4
8 – 3 =
32 + 4 3
– – 3 =
32 + 4 3 61

23. Write original equation.
GUIDED PRACTICE
for Examples 2, 3, and 4
Solve the equation.
5x2 = 80
SOLUTION
5x2 = 80
Write original equation.
x2 = 16
Divide each side by 5.
x = + 16
Take square roots of each side.
x = + 4
Product property
x =
+ 4
Simplify.

24. GUIDED PRACTICE
for Examples 2, 3, and 4
ANSWER
The solutions are and 4 – 4
Check Check the solutions by substituting them into the original equation.
5x2 = 80
5x2 = 80
5(4)2 = 80 ?
5(– 4)2 = 80 ?
5(16) = 80 ?
5(16) = 80 ?
80 = 80 
80 = 80 

25. Write original equation.
GUIDED PRACTICE
for Examples 2, 3, and 4
Solve the equation.
z2 – 7 = 29
SOLUTION
z2 – 7 = 29
Write original equation.
z2 = 36
Add 7 to each side.
z = + 36
Take square roots of each side.
z = + 6
Product property
z =
+ 6
Simplify.

26. GUIDED PRACTICE
for Examples 2, 3, and 4
ANSWER
The solutions are and 6 – 6
Check the solutions by substituting them into the original equation.
z2 – 7 = 29
z2 – 7 = 29
(6)2 – 7 = 29 ?
(– 6)2 – 7 = 29 ?
36 – 7 = 29 ? ?
36 – 7 = 29
29 = 29 
29 = 29 

27. Write original equation.
GUIDED PRACTICE
for Examples 2, 3, and 4
3(x – 2)2 = 40
SOLUTION
3(x – 2)2 = 40
Write original equation.
(x – 2)2 = 40 3
Divide each side by 3. +
(x – 2) = 40 3
Take square roots of each side.
x = 40 3 2 +
Division property
x = 40 3 2 + =
120 3 2 +

28. GUIDED PRACTICE
for Examples 2, 3, and 4 =
4(30) 3 2 + = 3 2 +
ANSWER
The solutions are 3 2 +

29. EXAMPLE 5
Model a dropped object with a quadratic function
Science Competition
For a science competition, students must design a container that prevents an egg from breaking when dropped from a height of 50 feet. How long does the container take to hit the ground ?

30. Substitute 0 for h and 50 for h0.
EXAMPLE 5
Model a dropped object with a quadratic function
SOLUTION
h = – 16t 2 + h0
Write height function.
0 = – 16t
Substitute 0 for h and 50 for h0.
– = – 16t 2
Subtract 50 from each side.
5016
= t2
Divide each side by – 16. 50 16 +
= t2
Take square roots of each side. t
Use a calculator.

31. EXAMPLE 5
Model a dropped object with a quadratic function
ANSWER
Reject the negative solution, – 1.8 , because time must be positive. The container will fall for about 1.8 seconds before it hits the ground.

32. Substitute 0 for h and 30 for h0.
GUIDED PRACTICE
GUIDED PRACTICE
for Example 5
What If? In Example 5, suppose the egg container is dropped from a height of 30 feet. How long does the container take to hit the ground?
SOLUTION
h = – 16t 2 + h0
Write height function.
0 = – 16t
Substitute 0 for h and 30 for h0.
– = – 16t 2
Subtract 30 from each side.
3016
= t2
Divide each side by – 16. 30 16 +
= t2
Take square roots of each side. t
Use a calculator.

33. GUIDED PRACTICE
GUIDED PRACTICE
for Example 5
ANSWER
Reject the negative solution, – 1.4, because time must be positive. The container will fall for about 1.4 seconds before it hits the ground.