Find each square root.

1. 6. –3.89 Homework 8.15 #1-7 Find each square root. 4. 3. 1. 2. 5. The area of a square piece of cloth is 68 in2. Estimate to the nearest tenth the side length of the cloth. Write all classifications that apply to each real number. 7.

2. 2 5 5 9 5 6 3 8 5 –1 Warm Up Simplify each expression. 1. 62 2. 112 121 36 25 36 81 4. 3. (–9)(–9) Write each fraction as a decimal. 0.4 5. 6. 0.5 –1.83 7. 5.375 8.

3. Vocabulary square root terminating decimal principal square root repeating decimal perfect square irrational numbers cube root natural numbers whole numbers integers rational numbers

4. A number that is multiplied by itself to form a product is a square root of that product. The radical symbol is used to represent square roots. For nonnegative numbers, the operations of squaring and finding a square root are inverse operations. In other words, for x ≥ 0, Positive real numbers have two square roots. Positive square root of 16 =4 4  4 = 42= 16 = –4 – (–4)(–4) = (–4)2= 16 Negative square root of 16

5. The principal square root of a number is the positive square root and is represented by . A negative square root is represented by – . The symbol is used to represent both square roots. A perfect square is a number whose positive square root is a whole number. Some examples of perfect squares are shown in the table. 0 1 4 9 16 25 36 49 64 81 100 02 12 22 32 42 52 62 72 82 92 102

6. Writing Math The small number to the left of the root is the index. In a square root, the index is understood to be 2. In other words, is the same as .

7. A number that is raised to the third power to form a product is a cube root of that product. The symbol indicates a cube root. Since 23 = 8, = 2. Similarly, the symbol indicates a fourth root: 24 = 16, so = 2.

8. Additional Example 1: Finding Roots Find each root. Think: What number squared equals 81? Think: What number squared equals 25?

9. Additional Example 1: Finding Roots Find the root. C. Think: What number cubed equals –216? (–6)(–6)(–6) = 36(–6) = –216 = –6

10. a. b. Partner Share! Example 1 Find each root. Think: What number squared equals 4? Think: What number squared equals 25?

11. Partner Share! Example 1 Find the root. c. Think: What number to the fourth power equals 81?

12. Think: What number squared equals Additional Example 2: Finding Roots of Fractions Find the root. A.

13. Think: What number cubed equals Additional Example 2: Finding Roots of Fractions Find the root. B.

14. Think: What number squared equals Additional Example 2: Finding Roots of Fractions Find the root. C.

15. Think: What number squared equals Partner Share! Example 2 Find the root. a.

16. Think: What number cubed equals Partner Share! Example 2 Find the root. b.

17. Think: What number squared equals Partner Share! Example 2c Find the root. c.

18. Square roots of numbers that are not perfect squares, such as 15, are not whole numbers. A calculator can approximate the value of as 3.872983346... Without a calculator, you can use square roots of perfect squares to help estimate the square roots of other numbers.

19. Since the area of the square is 13 in², then each side of the square is in. 13 is not a perfect square, so find two consecutive perfect squares that is between: 9 and 16. is between and , or 3 and 4. Refine the estimate. Additional Example 3: ArtApplication As part of her art project, Shonda will need to make a paper square covered in glitter. Her tube of glitter covers 13 in². Estimate to the nearest tenth the side length of a square with an area of 13 in².

20. Since 3.6 is too low and 3.65 is too high, is between 3.6 and 3.65. Round to the nearest tenth. The side length of the paper square is Additional Example 3 Continued 3.5 3.52 = 12.25 too low 3.6 3.62 = 12.96 too low 3.65 3.652 = 13.32 too high

21. Writing Math The symbol ≈ means “is approximately equal to.”

22. Since the area of the square is 26 ft², then each side of the square is ft. 26 is not a perfect square, so find two consecutive perfect squares that is between: 25 and 36. is between and , or 5 and 6. Refine the estimate. Partner Share! Example 3 What if…? Nancy decides to buy more wildflower seeds and now has enough to cover 26 ft2. Estimate to the nearest tenth the side length of a square garden with an area of 26 ft2.

23. Since 5.0 is too low and 5.1 is too high, is between 5.0 and 5.1. Rounded to the nearest tenth,  5.1. The side length of the square garden is  5.1 ft. Partner Share! Example 3 Continued 5.0 5.02 = 25 too low 5.1 5.12 = 26.01 too high

24. Reading Math Note the symbols for the sets of numbers. R: real numbers Q: rational numbers Z: integers W: whole numbers N: natural numbers

25. –32 can be written in the form . 14 is not a perfect square, so is irrational. Additional Example 4: Classifying Real Numbers Write all classifications that apply to each real number. A. –32 32 1 –32 = – –32 can be written as a terminating decimal. –32 = –32.0 rational number, integer, terminating decimal B. irrational

26. 7 can be written in the form . 67  9 = 7.444… = 7.4 49 can be written as a repeating decimal. –12 can be written in the form . Partner Share! Example 4 Write all classifications that apply to each real number. a. 7 rational number, repeating decimal b. –12 –12 can be written as a terminating decimal. rational number, terminating decimal, integer

27. 10 is not a perfect square, so is irrational. 100 is a perfect square, so is rational. 10 can be written in the form and as a terminating decimal. Partner Share! Example 4 Write all classifications that apply to each real number. irrational natural, rational, terminating decimal, whole, integer