2-F
Squares and square
roots of fractions
KEY CONCEPTS
It is possible to square and find the square root of fractions and mixed
numbers.
Squares and square roots of fractions can be modelled using area diagrams, as
shown below.
1
1
(1 2)2 = 2 4
22 = 4
1
( 2)2 =
12 = 1
1
4
1
( 3)2 =
1
9
Squares of fractions
To square a fraction, you multiply the fraction by itself.
A fraction can also be squared by squaring the numerator and then squaring
the denominator, as shown below.
2
 4
= 4 × 4 = 42 = 16
 7
7 7 7
49
2
To square a mixed number, write the mixed number as an improper fraction
and then square the fraction.
Calculate the value of
2
 5 2
 8 .
Calculate the value of 147 .
WRITE
WRITE
1
Write the question.
 4 2
17
52
82
2
Write the mixed
number as an improper
fraction.
=
11 2
 7 
25
64
3
Square the numerator
and square the
denominator.
=
112
72
=
121
49
Write the result as a
mixed number.
= 2 23
49
1
Write the question.
 5 2
 8
2
Square the numerator
and square the
denominator.
=
3
Simplify the result if
possible.
=
4
68
Maths XPRESS 7
Square roots of fractions
The square root of a fraction can be represented by the side length of a square;
the area of the square represents the value for which you are finding the square
root, as shown below.
4=2
214 = 112
1=1
1
4
= 12
1
9
The square root of a fraction can be obtained by finding the square root of the
numerator and the square root of the denominator separately, as shown at right.
If either of the numbers for which you are trying to find the square root is not a
perfect square, leave the number written as a square root;
for example,
36
97
=
36
97
=
= 13
81 =
169
81
9
=
13
169
6 .
97
EXAMPLE 4
Calculate the value of
36
121 .
Calculate the value of 1 79 .
WRITE
1
Write the question.
2
Find the square root
of the numerator
and the square root
of the denominator
separately.
=
Simplify the result
if possible.
=
3
36
121
36
121
6
11
WRITE
1
Write the question.
2
Write the mixed number as an
improper fraction.
=
16
9
3
Find the square root of the
numerator and the square root
of the denominator separately.
=
16
9
Write the result as a mixed
number.
= 1 13
4
1 79
=
4
3
LEARNING EXPERIENCE
Modelling squares and square roots of fractions
Equipment: BLM-23 1-cm grid paper, coloured pencils
1 Working individually, use 1-cm graph paper to draw a square representing 32.
2 Describe how your diagram can be used to show both 32 and 9.
3 Repeat step 1 to represent (2 12)2.
4 Use your diagram to estimate the value of (2 12)2.
5 Select two other mixed numbers between 1 and 3 and draw diagrams to represent and estimate the
value of their squares.
6 How could you use your graph paper to estimate the square root of a fraction?
Chapter 2
Fractions
69