1. Homework 8.15 #1-7
Find each square root. 3. 4. 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.
6. –3.89 7.

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

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. Positive real numbers have two square roots.
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.
4  4 = 42 = 16
= 4
Positive square
root of 16
(–4)(–4) = (–4)2 = 16 –
= –4
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. 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)(–6) = 36(–6) = –216

10. Partner Share! Example 1
Find each root. a.
Think: What number squared
equals 4? b.
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. Additional Example 2: Finding Roots of Fractions
Find the root. A.
Think: What number squared equals

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

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

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

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

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

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 Without a calculator, you can use square roots of perfect squares to help estimate the square roots of other numbers.

19. Additional Example 3: Art Application
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.
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 is between and , or 3 and 4. Refine the estimate.

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

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

22. 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.
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 is between and , or 5 and 6. Refine the estimate.

23. Partner Share! Example 3 Continued
= 25 too low
= too high
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.

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

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

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

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