This document provides a lesson on roots and irrational numbers. It begins with examples of simplifying expressions and writing fractions as decimals. It then defines key vocabulary like square root, principal square root, irrational numbers, and cube root. Several examples are provided of finding roots of numbers and fractions. The document explains that roots of non-perfect squares are irrational and cannot be expressed as terminating or repeating decimals. It also classifies different types of real numbers such as rational, irrational, integers, and natural numbers. Finally, it provides examples of classifying given real numbers and estimating the side length of a square based on its area.

1. 1-5 Roots and Irrational Numbers
Warm Up
Lesson Presentation
California Standards
Preview

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

3. 1-5 Roots and Irrational Numbers
2.0 Students understand and use such
operations as taking the opposite, finding the
reciprocal, taking a root, and raising to a
fractional power. They understand and use the
rules of exponents.
California
Standards

4. 1-5 Roots and Irrational Numbers
square root terminating decimal
principal square root repeating decimal
perfect square irrational numbers
cube root
natural numbers
whole numbers
integers
rational numbers
Vocabulary

5. 1-5 Roots and Irrational Numbers
4  4 = 42 = 16 = 4 Positive square
root of 16
(–4)(–4) = (–4)2 = 16 = –4 Negative square
root of 16
–
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.

6. 1-5 Roots and Irrational Numbers
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
02
1
12
100
4
22
9
32
16
42
25
52
36
62
49
72
64
82
81
92 102
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.

7. 1-5 Roots and Irrational Numbers
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 .
Writing Math

8. 1-5 Roots and Irrational Numbers
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: 2 = 16, so = 2.

9. 1-5 Roots and Irrational Numbers
Additional Example 1: Finding Roots
Find each root.
Think: What number squared equals 81?
Think: What number squared equals 25?

10. 1-5 Roots and Irrational Numbers
Find the root.
Think: What number cubed equals
–216?
Additional Example 1: Finding Roots
= –6 (–6)(–6)(–6) = 36(–6) = –216
C.

11. 1-5 Roots and Irrational Numbers
Find each root.
Check It Out! Example 1
Think: What number squared
equals 4?
Think: What number squared
equals 25?
a.
b.

12. 1-5 Roots and Irrational Numbers
Find the root.
Check It Out! Example 1
Think: What number to the fourth
power equals 81?
c.

13. 1-5 Roots and Irrational Numbers
Additional Example 2: Finding Roots of Fractions
Find the root.
Think: What number squared
equals
A.

14. 1-5 Roots and Irrational Numbers
Additional Example 2: Finding Roots of Fractions
Find the root.
Think: What number cubed equals
B.

15. 1-5 Roots and Irrational Numbers
Additional Example 2: Finding Roots of Fractions
Find the root.
Think: What number squared
equals
C.

16. 1-5 Roots and Irrational Numbers
Find the root.
Check It Out! Example 2
Think: What number squared
equals
a.

17. 1-5 Roots and Irrational Numbers
Find the root.
Check It Out! Example 2
Think: What number cubed
equals
b.

18. 1-5 Roots and Irrational Numbers
Find the root.
Check It Out! Example 2c
Think: What number squared
equals
c.

19. 1-5 Roots and Irrational Numbers
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.

20. 1-5 Roots and Irrational Numbers
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 16. is between
and , or 3 and 4. Refine the estimate.

21. 1-5 Roots and Irrational Numbers
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
The side length of the paper square is
Since 3.6 is too low and 3.65 is too high, is
between 3.6 and 3.65. Round to the nearest tenth.

22. 1-5 Roots and Irrational Numbers
The symbol ≈ means “is approximately equal to.”
Writing Math

23. 1-5 Roots and Irrational Numbers
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.
Check It Out! Example 3
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.

24. 1-5 Roots and Irrational Numbers
Check It Out! Example 3 Continued
5.0 5.02 = 25 too low
5.1 5.12 = 26.01 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.

25. 1-5 Roots and Irrational Numbers
Real numbers can be classified according to their
characteristics.
Natural numbers are the counting
numbers: 1, 2, 3, …
Whole numbers are the natural numbers
and zero: 0, 1, 2, 3, …
Integers are the whole numbers and their
opposites: –3, –2, –1, 0, 1, 2, 3, …

26. 1-5 Roots and Irrational Numbers
Rational numbers are numbers that can be
expressed in the form , where a and b are both
integers and b ≠ 0. When expressed as a
decimal, a rational number is either a terminating
decimal or a repeating decimal.
• A terminating decimal has a finite number of
digits after the decimal point (for example, 1.25,
2.75, and 4.0).
• A repeating decimal has a block of one or more
digits after the decimal point that repeat
continuously (where all digits are not zeros).

27. 1-5 Roots and Irrational Numbers
Irrational numbers are all numbers that are not
rational. They cannot be expressed in the form
where a and b are both integers and b ≠ 0. They
are neither terminating decimals nor repeating
decimals. For example:
0.10100100010000100000…
After the decimal point, this number contains 1
followed by one 0, and then 1 followed by two
0’s, and then 1 followed by three 0’s, and so on.
This decimal neither terminates nor repeats, so it is
an irrational number.

28. 1-5 Roots and Irrational Numbers
If a whole number is not a perfect square, then its
square root is irrational. For example, 2 is not a
perfect square and is irrational.

29. 1-5 Roots and Irrational Numbers
The real numbers are made up of all rational
and irrational numbers.

30. 1-5 Roots and Irrational Numbers
Note the symbols for the sets of numbers.
R: real numbers
Q: rational numbers
Z: integers
W: whole numbers
N: natural numbers
Reading Math

31. 1-5 Roots and Irrational Numbers
Additional Example 4: Classifying Real Numbers
Write all classifications that apply to each
real number.
A.
–32 = –
32
1
rational number, integer, terminating decimal
B.
irrational
–32
–32 can be written in the form .
14 is not a perfect square, so is
irrational.
–32 can be written as a terminating
decimal.
–32 = –32.0

32. 1-5 Roots and Irrational Numbers
Write all classifications that apply to each real
number.
a. 7
rational number, repeating decimal
Check It Out! Example 4
67  9 = 7.444… = 7.4
7 can be written in the form .
4
9
can be written as a repeating
decimal.
b. –12
–12 can be written in the form .
–12 can be written as a
terminating decimal.
rational number, terminating decimal, integer

33. 1-5 Roots and Irrational Numbers
Write all classifications that apply to each real
number.
Check It Out! Example 4
irrational
100 is a perfect square, so
is rational.
10 is not a perfect square, so
is irrational.
10 can be written in the form
and as a terminating decimal.
natural, rational, terminating decimal, whole, integer

34. 1-5 Roots and Irrational Numbers
Find each square root.
1. 2. 3. 4.
3
5. The area of a square piece of cloth is 68 in2.
Estimate to the nearest tenth the side length
of the cloth.  8.2 in.
Lesson Quiz
Write all classifications that apply to each
real number.
6. –3.89 7.
rational, repeating
decimal
irrational
1
5