Download square root

4-6 Squares and Square Roots
Preview
Evaluating Algebraic Expressions
Warm Up
Lesson Presentation
4-6 Squares and Square Roots
Warm
Up
Evaluating
Algebraic Expressions
Simplify.
1. 52
25
3. 122 144
5. 202 400
2. 82 64
4. 152 225
4-6 Squares and Square Roots
GPS
Evaluating Algebraic Expressions
M8N1. Students will understand
different representations of numbers
including
square roots, exponents, and scientific
notation.
4-6 Squares and Square Roots
GPS
Evaluating Algebraic Expressions
a. Find square roots of perfect squares.
b. Recognize the (positive) square root of a
number as a length of a side of a square with a
given area.
c. Recognize square roots as points and as
lengths on a number line.
4-6 Squares and Square Roots
GPS
Evaluating Algebraic Expressions
• d. Understand that the square root of 0 is 0
and that every positive number has two
square roots that are opposite in sign.
• e. Recognize and use the radical symbol to
denote the positive square root of a
4-6 Squares and Square Roots
Vocabulary
Evaluating
Algebraic Expressions
square root
principal square root
perfect square
radical
4-6 Squares and Square Roots
Because the area of a square can be expressed
using an exponent of 2, a number with an
Evaluating Algebraic Expressions
exponent of 2 is said to be squared. You read 32
as “three squared.”
3
Area = 32
3
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.
4-6 Squares and Square Roots
Evaluating Algebraic Expressions
Positive real numbers have two square
roots, one positive and one negative. The
positive square root, or principal square
root, is represented by
. The negative
square root is represented by –
.
4-6 Squares and Square Roots
Evaluating Algebraic Expressions
A perfect square is a number whose square roots
are integers. Some examples of perfect squares
are shown in the table.
4-6 Squares and Square Roots
Evaluating Algebraic Expressions
Writing Math
You can write the square roots of 16 as ±4,
which is read as “plus or minus four.”
4-6 Squares and Square Roots
Additional Example: 1 Finding the Positive and
Negative Square Roots of a Number
Evaluating
Algebraic
Expressions
Find
the two square
roots of each
number.
A. 49
49 = 7
–
7 is a square root, since 7 • 7 = 49.
49 = –7 –7 is also a square root, since –7 • (–7) = 49.
The square roots of 49 are ±7.
B. 100
100 = 10
– 100 = –10
10 is a square root, since 10 • 10 = 100.
–10 is also a square root, since
–10 • (–10) = 100.
The square roots of 100 are ±10.
4-6 Squares and Square Roots
Check It Out! Example 1
Find the two square roots of each number.
Evaluating
A. 25
–
Algebraic Expressions
25 = 5
5 is a square root, since 5 • 5 = 25.
25 = –5
–5 is also a square root, since –5 • (–5) = 25.
The square roots of 25 are ±5.
B. 144
144 = 12
12 is a square root, since 12 • 12 = 144.
– 144 = –12 –12 is also a square root, since
–12 • (–12) = 144.
The square roots of 144 are ±12.
4-6 Squares and Square Roots
Additional Example 2: Application
A Evaluating
square window
has an areaExpressions
of 169 square
Algebraic
inches. How wide is the window?
Find the square root of 169 to find the width of
the window. Use the positive square root; a
negative length has no meaning.
132 = 169
So
169 = 13.
The window is 13 inches wide.
4-6 Squares and Square Roots
Check It Out! Example 2
A square shaped kitchen table has an area of
Evaluating
Algebraic
Expressions
16
square feet. Will
it fit through
a van door
that has a 5 foot wide opening?
Find the square root of 16 to find the width of
the table. Use the positive square root; a
negative length has no meaning.
16 = 4
So the table is 4 feet wide, which is less than 5
feet, so it will fit through the van door.
4-6 Squares and Square Roots
Additional Example 3: Finding the Square Root
of a Monomial
Simplify
the expression.
Evaluating
Algebraic Expressions
A. 144c2
144c2 = (12c)2
= 12|c|
B.
Write the monomial as a square.
Use the absolute-value symbol.
z6
z6 = (z3)2
= |z3|
Write the monomial as a square:
z6 = (z3)2
Use the absolute-value symbol.
4-6 Squares and Square Roots
Additional Example 3: Finding the Square Root
of a Monomial
Evaluating Algebraic Expressions
Simplify the expression.
C. 100n4
100n4 =
(10n2)2 Write the monomial as a square.
= 10n2
10n2 is nonnegative for all
values of n. The absolutevalue symbol is not needed.
4-6 Squares and Square Roots
Check It Out! Example 3
Simplify
the expression.
Evaluating
Algebraic
Expressions
A. 121r2
121r2 = (11r)2
= 11|r|
B.
Write the monomial as a square.
Use the absolute-value symbol.
p8
p8 = (p4)2
= |p4|
Write the monomial as a square:
p8 = (p4)2
Use the absolute-value symbol.
4-6 Squares and Square Roots
Check It Out! Example 3
Simplify
the expression.
Evaluating
Algebraic
Expressions
C. 81m4
81m4 =
(9m2)2
= 9m2
Write the monomial as a square.
9m2 is nonnegative for all
values of m. The absolutevalue symbol is not needed.
4-6 Squares and Square Roots
Lesson Quiz
Find the two square roots of each number.
Evaluating Algebraic Expressions
1. 144
12
2. 2500 50
Simplify each expression.
3.
49p6
7|p3|
4.
z8
z4
5. Ms. Estefan wants to put a fence around 3 sides
of a square garden that has an area of 225 ft2.
How much fencing does she need?
45 ft
4-6 Squares and Square Roots
Rational Numbers
Evaluating Algebraic Expressions
A rational number is a number
that can be written as a ratio.
That means it can be written as
a fraction, in which both the
numerator (the number on top)
and the denominator (the
number on the bottom) are
whole numbers.
4-6 Squares and Square Roots
Irrational Numbers
Evaluating Algebraic Expressions
• Any indicated square root whose radicand is
not a perfect square is an irrational number.
• The numbers √
6, √
15, and √
• Most numbers that are not perfect squares
have square roots that are irrational
numbers
4-6 Squares and Square Roots
Evaluating Algebraic Expressions