Squaring Numbers and Finding Square Roots
The square of a number is found by multiplying the number by itself. An exponent of 2 is used to indicate that a number is being squared.
Example 1
Simplify.
-
5
2
5 squared
5
2
=
5
⋅
5
=
25
table with 2 rows and 3 columns , row1 column 1 , 5 squared , column 2 equals , column 3 5 dot 5 , row2 column 1 , , column 2 equals , column 3 25 , end table
-
(
−
3.5
)
2
open minus . 3.5 , close squared
(
−
3.5
)
2
=
(
−
3.5
)
⋅
(
−
3.5
)
=
12.25
table with 2 rows and 3 columns , row1 column 1 , open negative 3.5 close squared , column 2 equals , column 3 open negative 3.5 close dot open negative 3.5 close , row2 column 1 , , column 2 equals , column 3 12.25 , end table
-
(
2
7
)
2
open , 2 sevenths , close squared
(
2
7
)
2
=
2
7
⋅
2
7
=
4
49
table with 2 rows and 1 column , row1 column 1 , open , 2 sevenths , close squared . equals , 2 sevenths , dot , 2 sevenths , row2 column 1 , equals , 4 forty ninths , end table
The square root of a number is itself a number that, when squared, results in the original number. A radical symbol
(
)
open square root of end root close is used to represent the positive square root of a number.
Example 2
Simplify. Round to the nearest tenth if necessary.
-
36
36
=
6
,
since
6
2
=
36
.
table with 2 rows and 1 column , row1 column 1 , square root of 36 , row2 column 1 , square root of 36 equals 6 comma , since . 6 squared , equals 36 . , end table
-
174
174
=
13.2
,
since
13.2
2
≈
174
.
table with 2 rows and 1 column , row1 column 1 , square root of 174 , row2 column 1 , square root of 174 , equals , 13.2 , comma , since . 13.2 squared , almost equal to 174 . , end table
You can solve equations that include squared numbers.
Example 3
Algebra Solve.
-
x
2
=
144
x
=
12
or
−
12
table with 2 rows and 3 columns , row1 column 1 , bold italic x squared , column 2 equals , column 3 144 , row2 column 1 , x , column 2 equals , column 3 12 , or , minus 12 , end table
-
a
2
+
3
2
=
5
2
a
2
+
9
=
25
a
2
=
16
a
=
4
or
−
4
table with 4 rows and 3 columns , row1 column 1 , bold italic eh squared , plus , 3 squared , column 2 equals , column 3 5 squared , row2 column 1 , eh squared , plus 9 , column 2 equals , column 3 25 , row3 column 1 , eh squared , column 2 equals , column 3 16 , row4 column 1 , eh , column 2 equals , column 3 4 , or , minus 4 , end table
Exercises
Simplify.
-
11
2
11 squared
-
(
−
14
)
2
open minus . 14 , close squared
-
5
.
1
2
5 . , 1 squared
-
(
8
5
)
2
open , 8 fifths , close squared
-
−
6
2
negative , 6 squared
-
(
−
3
7
)
2
open negative , 3 sevenths , close squared
Simplify. Round to the nearest tenth if necessary.
-
100
square root of 100
-
169
square root of 169
-
74
square root of 74
-
50
square root of 50
-
4
9
square root of 4 ninths end root
-
49
81
square root of 49 over 81 end root
Algebra Solve. Round to the nearest tenth if necessary.
-
x
2
=
49
x squared , equals 49
-
a
2
=
9
eh squared , equals 9
-
y
2
+
7
=
8
y squared , plus , 7 equals 8
-
5
+
x
2
=
11
5 plus , x squared , equals 11
-
8
2
+
b
2
=
10
2
8 squared , plus , b squared , equals , 10 squared
-
5
2
+
4
2
=
c
2
5 squared , plus , 4 squared , equals , c squared
-
p
2
+
12
2
=
13
2
p squared , plus , 12 squared , equals , 13 squared
-
20
2
=
15
2
+
a
2
20 squared , equals , 15 squared , plus , eh squared
