# Square Roots Real Numbers

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```					Simplify each expression.

1.   256     16             2.    361    19

3.   625      25            4.    400     20

Tell which two whole numbers each square root falls
between.
5.    30 5 and 6           6.   5     2 and 3

7.   12    3 and 4          8.    99    9 and 10
Vocabulary, Rational, Irrational, Classifying
Evaluating Expressions Involving Square Roots

Evaluate the expression.
3 36 + 7

3 36 + 7 = 3(6) + 7     Evaluate the square root.

= 18 + 7      Multiply.

Try This!
Evaluate the expression.

2   25 + 4

2 25 + 4 = 2(5) + 4           Evaluate the square root.

= 10 + 4         Multiply.

Natural numbers are the counting numbers: 1, 2, 3, …

Whole numbers are the natural numbers and zero: 0, 1, 2, 3, …

Integers are whole numbers and their opposites: –3, –2, –1, 0, 1,
2, 3, …

a
Rational numbers can be expressed in the form b ,
1 7 9
where a and b are both integers and b ≠ 0: , ,
2 1 10 .
Terminating decimals are rational numbers in decimal form
that have a finite number of digits: 1.5, 2.75, 4.0

Repeating decimals are rational numbers in decimal form
that have a block of one or more digits that repeat
continuously: 1.3, 0.6, 2.14
a
Irrational numbers cannot be expressed in the form b . They
include square roots of whole numbers that are not perfect
squares and nonterminating decimals that do not repeat:
,     , 
Real Numbers
Rational Numbers (Q)   Irrational
Numbers
Integers (Z)

Whole Numbers (W)

Natural Numbers (N)
The square roots of many numbers like        , 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.
Estimating Square Roots of Numbers

Each square root is between two integers. Name the

Think: What are perfect
55           squares close to 55?
72 = 49        49 < 55

82 = 64        64 > 55

55 is between 7 and 8 because 55 is between 49 and
64.
Estimating Square Roots of Numbers

Each square root is between two integers. Name the
Think: What are perfect
–   90
squares close to 90?
–92 = 81     81 < 90

–102 = 100       100 > 90

–    90 is between –9 and –10 because 90 is between 81
and 100.
TRY THIS!
Each square root is between two integers. Name the
Think: What are perfect
80
squares close to 80?
82 = 64        64 < 80

92 = 81        81 > 80

80 is between 8 and 9 because 80 is between 64 and 81.
Try This!

Each square root is between two integers. Name the
integers.

–               Think: What are perfect
45
squares close to 45?
–62 = 36            36 < 45

–72 = 49            49 > 45

–       45 is between –6 and –7 because 45 is between 36 and
49.
Problem-Solving Application

As part of her art project, Shonda will need to make a
square covered in glitter. Her tube of glitter covers 13
square inches. What is the greatest side length
Shonda’s square can have?

1
Understand the problem

The answer will be the side length of the square.
List the important information:
• The tube of glitter can cover an area of 13 square inches.
Continued

2   Make a Plan

The side length of the square is    because
      = 13. Because 13 is not a perfect
square,       is not a whole number. Estimate
to the nearest tenth.

Find the two whole numbers that        is between.
Because 13 is between the perfect squares 9 and
16.        is between     and        , or between
3 and 4.
Estimating Real Numbers

Because 13 is closer to 16 than to 9,   is
closer to 4 than to 3.

3                     4

You can use a guess-and-check method to estimate .   .
Estimating Real Numbers

3       Solve

Guess 3.6: 3.62 = 12.96 too low                is greater than 3.6.

Guess 3.7: 3.72 = 13.69 too high              is less than 3.7.

3                        3.6       3.7                       4
Because 13 is closer to 12.96 than to 13.69,             is closer to
3.6 than to 3.7.           3.6
Continued

4   Look Back

A square with a side length of 3.6 inches
would have an area of 12.96 square inches.
Because 12.96 is close to 13, 3.6 inches
is a reasonable estimate.
Try This!

What if…? Nancy decides to buy more wildflower seeds and now
has enough to cover 38 ft2. What is the side length of a square
garden with an area of 38 ft2 to the nearest tenth?

Use a guess and check method to estimate            .

Guess 6.1 6.12 = 37.21 too low                      is greater than 6.1.

Guess 6.2 6.22 = 38.44 too high                    is less than 6.2.

A square garden with a side length of 6.2 ft would have an
area of 38.44 ft2. 38.44 ft is close to 38, so 6.2 is a reasonable
All numbers that can be represented on a number
line are called real numbers and can be classified
according to their characteristics.
Classifying Real Numbers
Write all classifications that apply to each Real number.
A. –32
32 can be written as a fraction and a
–32 = – 32 = –32.0           decimal.
1
rational number, integer, terminating decimal

B. 5
5                        5 can be written as a fraction and a
5 = 1 = 5.0                  decimal.
rational number, integer, whole number, natural number, terminating
decimal
C.     69                    The digits continue with no pattern.
69  8.306623863
irrational number
Classifying Real Numbers
Write all classifications that apply to each Real number.

A.       9         9 =3
rational number, integer, whole number, natural number, terminating
decimal
B.   –35.9        –35.9 is a terminating decimal.
rational number, terminating decimal

81              81        9
C.                         =     =3
3                3        3
rational number, integer, whole number, natural number, terminating
decimal
Try This!
Write all classifications that apply to each real number.

4
4
3a.   7
7 can be written as a repeating
9
9                decimal.
67  9 = 7.444… = 7.4
rational number, repeating decimal
3b.   –12
-12 can be written as a fraction
–12 = – 12 = –12.0      and a decimal.
1
rational number, terminating decimal, integer
3c.
The digits continue with no pattern.
= 3.16227766…
irrational number
Lesson Quiz

Find each square root.
2.            -8            3             1
1.         12                         3.          4.   –
7             2
5. The area of a square piece of cloth is 68 in2.
How long is each side of the piece of cloth?
inch. 8.2 in.
Write all classifications that apply to each real number.
6. 1 rational, integer, whole number, natural number,
terminating decimal

7. –3.89        rational, repeating decimal

8.              irrational

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