1. Preview
Warm Up
California Standards
Lesson Presentation

2. Warm Up
Simplify. 25 64
144
225
5. 202
400

3. NS2.4 Use the inverse relationship between raising to a power and extracting the root of a perfect square integer; for an integer that is not a square, determine without a calculator the two integers between which its square root lies and explain why.
Also covered: AF2.2
California
Standards

4. Vocabulary
square root
principal square root
perfect square

5. Because the area of a square can be expressed using an exponent of 2, a number with an exponent of 2 is said to be squared. You read 32 as “three squared.” 3
Area = 32
The square root of a number is one of the two equal factors of that number. Squaring a nonnegative number and finding the square root of that number are inverse operations.

6. Positive real numbers have two square roots, one positive and one negative. The positive square root, or principle square root, is represented by The negative square root is represented by –

7. A perfect square is a number whose square roots are integers
A perfect square is a number whose square roots are integers. Some examples of perfect squares are shown in the table.

8. You can write the square roots of 16 as ±4, which is read as “plus or minus four.”
Writing Math

9. Additional Example: 1 Finding the Positive and Negative Square Roots of a Number
Find the two square roots of each number.
A. 49
49 = 7
7 is a square root, since 7 • 7 = 49.
49 = –7 –
–7 is also a square root, since –7 • (–7) = 49.
The square roots of 49 are ±7.
B. 100
100 = 10
10 is a square root, since 10 • 10 = 100.
100 = –10 –
–10 is also a square root, since
–10 • (–10) = 100.
The square roots of 100 are ±10.

10. Check It Out! Example 1
Find the two square roots of each number.
A. 25
25 = 5
5 is a square root, since 5 • 5 = 25.
25 = –5 –
–5 is also a square root, since –5 • (–5) = 25.
The square roots of 25 are ±5.
B. 144
144 = 12
12 is a square root, since 12 • 12 = 144.
144 = –12 –
–12 is also a square root, since
–12 • (–12) = 144.
The square roots of 144 are ±12.

11. Additional Example 2: Application
A square window has an area of 169 square inches. How wide is the window?
Find the square root of 169 to find the width of the window. Use the positive square root; a negative length has no meaning.
132 = 169
So = 13.
The window is 13 inches wide.

12. Check It Out! Example 2
A square shaped kitchen table has an area of 16 square feet. Will it fit through a van door that has a 5 foot wide opening?
Find the square root of 16 to find the width of the table. Use the positive square root; a negative length has no meaning.
16 = 4
So the table is 4 feet wide, which is less than 5 feet, so it will fit through the van door.

13. Additional Example 3: Finding the Square Root of a Monomial
Simplify the expression. A.
144c2
144c2 = (12c)2
Write the monomial as a square.
= 12|c|
Use the absolute-value symbol. B. z6
Write the monomial as a square: z6 = (z3)2
z6 = (z3)2
= |z3|
Use the absolute-value symbol.

14. Additional Example 3: Finding the Square Root of a Monomial
Simplify the expression. C.
100n4
100n4 = (10n2)2
Write the monomial as a square.
= 10n2
10n2 is nonnegative for all values of n. The absolute-value symbol is not needed.

15. Check It Out! Example 3
Simplify the expression. A.
121r2
121r2 = (11r)2
Write the monomial as a square.
= 11|r|
Use the absolute-value symbol. B. p8
Write the monomial as a square: p8 = (p4)2
p8 = (p4)2
= |p4|
Use the absolute-value symbol.

16. Check It Out! Example 3
Simplify the expression. C.
81m4
81m4 = (9m2)2
Write the monomial as a square.
= 9m2
9m2 is nonnegative for all values of m. The absolute-value symbol is not needed.

17. Lesson Quiz
Find the two square roots of each number.
Simplify each expression.
p z8
±12
±50
7|p3| z4
5. Ms. Estefan wants to put a fence around 3 sides of a square garden that has an area of 225 ft2. How much fencing does she need?
45 ft