Warm Up
Simplify each expression.
1. 62
36
3. (–9)(–9) 81
2. 112 121
4.
25
36
Write each fraction as a decimal.
5. 2 0.4
6. 5
0.5
5
9
7. 5 3 5.375
8
8. –1 5
6
–1.83
Roots and
Irrational Numbers
Section 1.5
Objectives:
In this lesson you’ll:
• Evaluate expressions containing roots.
• Classify numbers within the real number system
Words to know…
• Square root - a number which, when multiplied by itself,
produces the given number. (Ex. 7² = 49, 7 is the square
• root of 49)
• Perfect square- any number that has an integer square
root.(ex. 100 is a perfect square ,
100  10
• Cube root - a number that is raised to the third power to form
a product is a cube root. (ex 23=8,
=2)
Square Roots
Squares
Perfect Square Roots
0² = 0
0 0
1² = 1
4² = 16
1 1
4 2
9 3
16  4
5² = 25
25  5
6² = 36
36  6
7² = 49
49  7
8² = 64
64  8
2² = 4
3² = 9
9² = 81
10² = 100
Are squares and square roots inverses?
3 9
9 3
5  25
25  5
9  81
81  9
2
2
2
A square root is the inverse operation of a square!
Do you know your perfect squares?
1) 49  ? 7 and -7
4)5  ?
25
2) 64  ? 8 and -8
5)11  ?
121
3) 9  ? 3and -3
6)14  ? 196

2
2
2
Square Roots
Positive real numbers have two square roots.
Find the square roots of 16.
4  4 = 42 = 16
(–4)(–4) = (–4)2 = 16
–
=4
Positive square
root of 16
= –4
Negative square
root of 16
The square roots of 16 are 4 and - 4.
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
.
Cube roots
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.
You try
Find each root.
Think: What number squared equals 81?
Think: What number squared equals 25?
C.
Think: What number cubed equals –216?
= –6
(–6)(–6)(–6) = 36(–6) = –216
You try
Finding Roots of Fractions.
a.
Think: What number squared
equals
b.
Think: What number cubed
equals
You try
Finding Roots of Fractions.
Think: What number squared
A.
equals
B.
Think: What number cubed equals
Approximating Square Roots
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.
Remember
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.
Approximating Square Roots
Approximate
54 to the nearest whole number.
7  49
2
54
8  64
2
Is between 7² and 8².
54  7.3
Let’s practice…
Determine what two consecutive integers each root lies between.
8
17 
24 
35 
Between 2 and 3
Between 4 and 5
Between 4 and 5
Between 5 and 6
Words to know…
• Natural numbers - The counting numbers. (example: 1, 2,
3…)
• Whole numbers - The natural numbers and zero.(example: 0,
1,2,3…)
• Integers -The whole numbers and their opposites.(ex: …-3,-2,1,0,1,2,3…)
• Rational numbers - Numbers that can be expressed as a
fraction (a/b).
Words to know…
• Terminating decimal -Rational numbers in decimal form that
have finite (ends) number of digits. (ex 2/5= 0.40 )
• Repeating decimal -rational numbers in decimal form that
have a block for one or more digits that repeats continuously.
(ex. 1.3=1.333333333)
• Irrational numbers - numbers that cannot be expressed as a
fraction including square roots of whole numbers that are not
perfect squares and nonterminating decimals that do not
repeat.
The real numbers are made up of all rational
and irrational numbers.
Reading Math
Note the symbols for the sets of numbers.
R: real numbers
Q: rational numbers
Z: integers
W: whole numbers
N: natural numbers
Classifying Real Numbers
Write all classifications that apply to each
real number.
A. –32
32
–32 = –
1
–32 = –32.0
–32 can be written in the form
.
–32 can be written as a terminating
decimal.
rational number, integer, terminating decimal
B.
irrational
14 is not a perfect square, so
irrational.
is
Check It Out!
Write all classifications that apply to each real
number.
a. 7
7 4 can be written in the form
9
.
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
.
Write all classifications that apply to each real
number.
irrational
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.
natural, rational, terminating decimal, whole, integer
A challenge…
• Would you know how to solve this….
x  11  36
2
x  25
2
x  25
2
A challenge…
• Solve the variable.
x  3  67
2
x  64
2
x  64
2
Lesson Quiz
Find each square root.
1.
3
2.
3.
5
4.
1
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.
Write all classifications that apply to each real
number.
6. –3.89 rational, repeating
decimal
7.
irrational