The document provides examples and explanations for finding square roots of numbers and monomial expressions. It discusses perfect squares, principal square roots, and using the inverse relationship between exponents and roots. Examples are provided for finding the positive and negative square roots of numbers, evaluating square roots of areas of squares, and simplifying square root expressions involving monomials.

1. Warm Up California Standards Lesson Presentation Preview

2. Warm Up Simplify. 25 64 144 225 400 1. 5 2 2. 8 2 3. 12 2 4. 15 2 5. 20 2

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. 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. 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 3 2 as “three squared.” 3 3 Area = 3 2

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. 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. 7 is a square root, since 7 • 7 = 49. – 7 is also a square root, since –7 • (–7) = 49. 10 is a square root, since 10 • 10 = 100. – 10 is also a square root, since – 10 • (–10) = 100. A. 49 B. 100 The square roots of 49 are ±7. The square roots of 100 are ±10. 49 = –7 – 49 = 7 100 = 10 100 = –10 –

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

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

12. Find the square root of 16 to find the width of the table. Use the positive square root; a negative length has no meaning. 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? So the table is 4 feet wide, which is less than 5 feet, so it will fit through the van door. 16 = 4

13. Additional Example 3: Finding the Square Root of a Monomial Simplify the expression. A. Write the monomial as a square. Use the absolute-value symbol. = 12| c | 144 c 2 B. z 6 = | z 3 | Write the monomial as a square: z 6 = (z 3 ) 2 Use the absolute-value symbol. 144 c 2 = (12 c ) 2 z 6 = ( z 3 ) 2

14. Additional Example 3: Finding the Square Root of a Monomial Simplify the expression. C. Write the monomial as a square. 10n 2 is nonnegative for all values of n. The absolute-value symbol is not needed. = 10 n 2 100 n 4 100 n 4 = (10 n 2 ) 2

15. Check It Out! Example 3 Simplify the expression. A. Write the monomial as a square. Use the absolute-value symbol. = 11| r | 121 r 2 B. p 8 = | p 4 | Write the monomial as a square: p 8 = (p 4 ) 2 Use the absolute-value symbol. 121 r 2 = (11 r ) 2 p 8 = ( p 4 ) 2

16. Check It Out! Example 3 Simplify the expression. C. Write the monomial as a square. 9m 2 is nonnegative for all values of m. The absolute-value symbol is not needed. = 9 m 2 81 m 4 81 m 4 = (9 m 2 ) 2

17. Lesson Quiz  12  50 7| p 3 | z 4 5. Ms. Estefan wants to put a fence around 3 sides of a square garden that has an area of 225 ft 2 . How much fencing does she need? 45 ft Find the two square roots of each number. 1. 144 2. 2500 Simplify each expression. 3. 49 p 6 4. z 8