Download Squares, square roots, cubes and cube roots

TOPIC 2
Squares, square roots, cubes and
cube roots
By the end of this topic, you should be able to:
ü Find squares, square roots, cubes and cube roots of positive whole
numbers, decimals and common fractions
ü Use a calculator to find squares, square roots, cubes and cube roots
of any number.
Oral activity
Discuss squares, square roots, cubes
and cube roots
PAIR
1 Discuss what you remember about squares, square roots,
cubes and cube roots.
2 Give an example of each and compare your examples with
those of your partner.
4
x2
2ndF
__
1
√
0
0 yx
__
√
4
2ndF √
__
yx 3
x2
1 2 5 You can use a calculator to find squares, square roots, cubes and cube roots.
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TOPIC 2 Squares, square roots, cubes and cube roots
Square numbers and cube numbers
Key concept
To find a square number, multiply a number by itself. For example,
16 is a square number because 4 4 = 16. To find a cube number,
multiply a number by itself twice. For example, 27 is a cube number
because 3 3 3 = 27.
Activity 1
Identify square numbers and
cube numbers
Learn that square
numbers and cube
numbers are special
cases of whole
numbers.
SINGLE
1 Give examples of squares, square roots, cubes and cube roots.
2 Draw a number square from 1 to 100 in your exercise book.
3 Draw a box around each square number in your
number square.
a) Write all the square numbers as a number sequence.
b) Count the number of members between each pair of
square numbers. Check that these numbers form a
number sequence.
c) Use the rule of this number sequence to find the next
two square numbers in the sequence. Find a way to
check your answers.
4 Draw a number square from 1 to 100 in your exercise
book. Draw a circle around each cube number in your
number square.
a) Write all the cube numbers as a number sequence.
b) Write the next three numbers in this sequence.
Square numbers
Key concept
Squares and square roots of whole numbers
Determine the square
roots of numbers.
We also use index notation to write square numbers. For example,
9 9 = 92 and we read 92 as “9 squared”.
If you multiply a given number by itself to get a square number,
that given number is a square root. For example, the square roots
of 16 are __
4 and 4 because 4 4 = 16
4 = 16. We use the
___ and 4 ___
symbol √ for “square root”, so √ 16 = 4 and ___
√ 16 = 4. If there is
no or sign, assume a sign, for example, √ 25 = 5.
Example 1
Use mental maths, pen-and-paper methods or a calculator to find
the square of each number.
1 5
2 9
3 13
4 26
5 37
Solutions
1 25
2 81
3 169
4 676
5 1 369
SO 2.1.9.1.1; 2.1.9.1.2; 2.1.10.1.2
17
TOPIC 2 Squares, square roots, cubes and cube roots
Example 2
Use a calculator to find the square of 214.
Solution
The key sequence for a simple calculator is
2 1 4
2 1 4
The key sequence for a calculator with an x2 function is
2 1 4
x2
2142 = 45 796
Exercise 1
Find squares and square roots of numbers
1 Find the square of each number.
a) 1
b) 2
c) 3
d) 4
e) 6
2 Use mental maths or pen-and-paper methods to find the
square of each number.
a) 7
b) 8
c) 12
d) 10
e) 11
3 Use a calculator to find the square of each number.
a) 29
b) 52
c) 88
d) 277
e) 3 914
4 Find the square root of each number.
a) 1
b) 4
c) 9
d) 16
e) 25
5 Use mental maths to find the square root of each number.
a) 64
b) 100 c) 49
d) 81
e) 121 f) 25
6 Find the
each equation.
___
___
__ missing number in
=…
a) √ 4 ____
b) √ ____
25 = …
c) ___
√ 64 = …
d) √ 100 = …
e) √ 144 = …
f) √ 49 = …
SINGLE
Squares and square roots of common fractions
To find the square of a common fraction, multiply the common
fraction by itself. For example __12 __12 or ( __12 )2 = __14 . To multiply two
fractions, multiply the numerators with each other and the
denominators with each other.
To find the square root of a common fraction, find the square root of
the numerator and the denominator.
Example 3
18
1 Find the square of __23.
9.
2 Find the square root of __
64
Solutions
1 __23 __23 = __49
2
SO 2.1.9.1.1; 2.1.9.1.2
___
3
9 = __
__
√ 64
8
TOPIC 2 Squares, square roots, cubes and cube roots
Exercise 2
Find squares and square roots of
common fractions
SINGLE
1 Find the square of each fraction.
b) __34
c) __35
d) __27
e) __58
a) __25
2 Find the missing number in each equation.
b) ( __12 )2 = …
c) ( __49 )2 = …
a) ( __38 )2 = …
d) ( __15 )2 = …
e) ( __27 )2 = …
f) ( __35 )2 = …
3 Find the square root of each fraction.
9
25
4
b) __14
c) __
d) __
e) __
a) __49
16
25
36
4 Find the
missing
number
in
each
equation.
___
__
__
16 = …
b) __19 = …
c) __
a) __14 = …
49
d)
√ ___
9 =…
64
√ __
e)
√ ___
25 = …
81
√ __
f)
√ ___
49 = …
25
√ __
Squares and square roots of decimals
Note
To find the square root of a common fraction that is not a perfect
square, convert it to a decimal fraction.
Then
__
_____ use a calculator with a
square root function, for example __23 √ 0.67 0.82.
When a question
does not state to how
many decimal places
you must round
the answer, round
the answer to two
decimal places.
√
Estimate the
__ answer so that you can check the calculation. For
example,
√ 5 should be between 2 and 3 because 22 = 4 and 32 = 9.
__
So √ 5 = 2.24 makes sense.
Example 4
Find the square of each decimal correct to two decimal places,
where necessary.
1 0.16
2 0.31
3 5.8
4 34.89
5 175.09
Solutions
1 0.03
4 1 217.31
2 0.10
5 30 656.51
3 33.64
Example 5
Estimate the square root of each decimal correct to two decimal
places. Use a calculator to find the square root, then use your
estimates to check the answers.
_____
____
_____
______
______
1 √ 0.09
2 √ 1.8
3 √ 15.4
4 √ 37.45 5 √ 357.2
Solutions
1 0.3
2
1.34
3 3.92
4
6.12
5 18.90
SO 2.1.9.1.1; 2.1.9.1.2
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TOPIC 2 Squares, square roots, cubes and cube roots
Exercise 3
Find squares and square roots of decimals
1 Use a calculator to find the square of each decimal correct
to two decimal places.
a) 0.8
b) 0.9
c) 1.4
d) 2.9
e) 3.3
2 Use a calculator to find the square of each decimal correct
to two decimal places, where necessary.
a) 1.56
b) 2.83
c) 13.55 d) 38.19 e) 42.72
3 Find the missing number in each case. Give the answers
correct to two decimal places, where necessary.
a) 3.42 = …
b) 17.942 = …
c) 44.052 = …
d) 57.592 = …
e) 92.652 = …
f) 126.332 = …
4 Estimate the square root in each case. Then use a calculator
to find the square root correct to two decimal places.
a) 9.8
b) 17.64 c) 24.08 d) 50.44 e) 67.32
5 Find the missing number in each case correct to two
decimal places.
_____
______
______
=
=
a) √ ______
4.56 = …
b) √ 10.31
c) √ 35.18
______ …
_______ …
d) √ 79.29 = …
e) √ 105.8 = …
f) √ 187.52 = …
Key concept
Cube numbers
Determine the cube
roots of numbers.
Cubes and cube roots of whole numbers and decimals
SINGLE
We also use index notation to write cube numbers. For example,
2 2 2 = 23 and we read 23 as “2 to the power of 3”.
If you multiply a given number by itself twice to get a cube number,
that given number is a cube root. For example, __the cube root of 27
3
is 3 ___
because 3 ___
3 3 = 27. We use the symbol √ for “cube root”,
3
3
so √64 = 4 and √ 43 = 4.
You need a calculator with 2ndF and y x keys to find cubes and cube
roots in difficult cases. You can also write numbers as products of
their prime factors to help you find cube roots.
Example 6
Use a calculator to find the cube of 11.3.
Solution
The key sequence for a simple calculator is
1 1
.
3
1 1
.
3
1 1
.
3
The key sequence for a calculator with a y x function is
1 1
.
3
yx 3
11.33 = 1 442.90
20
SO 2.1.9.1.1; 2.1.9.1.2; 2.1.10.1.2
TOPIC 2 Squares, square roots, cubes and cube roots
Example 7
Write 216 as the product of its prime factors, then find its cube root.
Solution
2 216
Divide 216 by 2 and write 108 on the next line.
2 108
Divide 108 by 2 and write 54 on the next line.
2 54
Divide 54 by 2 and write 27 on the next line.
3 27
Divide 27 by 3 and write 9 on the next line.
3 9
Divide 9 by 3 and write 3 on the next line.
3 1
216 = 2 _______
2 2 3 3 3 = 23 33
____ 3
3
√ 216 = √ 23 33 = 2 3 = 6
Note
Example 8
Estimate the cube root of 37.8 and explain your decision. Use a
calculator to find the value correct to two decimal places. Use your
estimate to check the answer.
Estimating cube roots
helps us to check
calculator answers.
Solution
_____
3
Estimate 3.2 because √ 37.8 should be between 3 and 4. 33 = 27
and 43 = 64, so the answer is closer to 3 than 4 because 37.8 is
closer to 27 than to 64.
The key sequence is
3 2ndF
3
yx 3 7 . 8
_____
√ 37.8 = 3.36
The answer of 3.36 makes sense because it is close to the estimate.
Exercise 4
Emerging issue
Find cubes and cube roots of whole
numbers and decimals
SINGLE
1 Use pen-and-paper methods to find the cube of each
number.
a) 7
b) 9
c) 13
d) 18
e) 20
2 Use a calculator to find the cube of each decimal correct to
two decimal places.
a) 1.4
b) 2.7
c) 6.3
d) 13.1
e) 26.89
3 Use a calculator to find the missing number in each case
correct to two decimal places.
a) 1.33 = …
b) 2.13 = …
c) 3.93 = …
3
3
d) 4.15 = …
e) 6.08 = …
f) 10.583 = …
In engineering, we
use square roots to
calculate the radii of
circles and the sides
of squares. We also
use cube roots to
calculate the radii of
spheres and the sides
of cubes.
SO 2.1.9.1.1; 2.1.9.1.2; 2.1.10.1.2
21
TOPIC 2 Squares, square roots, cubes and cube roots
4 Write each number as the product of its prime factors.
Then find the cube root of each number.
a) 27
b) 64
c) 125
d) 343
e) 729
5 Estimate the cube root of each decimal, then use a
calculator to find the value correct to two decimal places.
Use your estimates to check the answers.
a) 1.4
b) 7.5
c) 54.9
d) 120
e) 201.4
Cubes and cube roots of common fractions
To find the cube of a common fraction, multiply the common
fraction by itself twice. For example, __12 __12 __12 or ( __12 )3 = __18 . To multiply
fractions, multiply the numerators with one another and the
denominators with one another.
To find the cube root of a common fraction, find the cube root of the
numerator and the denominator.
Example 9
1 Find the cube of __25 .
27 .
2 Find the cube root of ___
125
Solutions
8
1 __25 __25 __25 = ___
125
2
3
___
3
27 = __
___
√ 125
5
Exercise 5
Find cubes and cube roots of
common fractions
1 Find the cube of each fraction.
a) __12
b) __13
c) __14
d) __15
2 Find the cube of each fraction.
a) __23
b) __34
c) __35
d) __27
3 Find the missing number in each case.
b) ( __13 )3 = …
c)
a) ( __12 )3 = …
d) ( __56 )3 = …
e) ( __38 )3 = …
f)
4 Find the cube root of each fraction.
8
1
27
a) __18
b) __
c) __
d) __
27
27
64
5 Find the
missing number in each case.
___
___
3
3
8 =…
1 =…
___
b)
a) __
64
125
c)
e)
22
√ ___
1 =…
216
√___
___
64 = …
125
√___
3
3
SO 2.1.9.1.1; 2.1.9.1.2; 2.1.10.1.2
d)
f)
√ ___
64 = …
27
√__
___
27 = …
8
√ __
3
3
SINGLE
e) __16
e) __58
( __15 )3 = …
( __24 )3 = …
125
e) ___
216
TOPIC 2 Squares, square roots, cubes and cube roots
Summary
ü Multiply a number by itself to get the
square of the number. For example, 25 is
the square of 5 because 5 5 = 25.
ü The square root of a number p is the
number that you have to multiply
by
___
itself to get p. For example, √ 36 = 6
because 6 6 = 36.
ü Multiply a number by itself twice to get
the cube of the number. For example, 64
is the cube of 4 because 4 4 4 = 64.
ü The cube root of a number q is the
number that you have to multiply ____
by
3
itself twice to get q. For example, √343 = 7
because 7 7 7 = 343.
Revision
1 Write only the answer for each
expression correct to two decimal places,
where necessary.
a) 52
b) 1.52
2
c) 0.3
d) 0.43
e) ( __32 )2
f ) 33
2 Estimate the value of each expression,
then use a calculator to find the value.
Give the answers correct to two decimal
places, where necessary.
a) 113
b) ( __53 )2
d) 2.913
c) 1.442
e) 3.363
f ) 10.032
3 Estimate the square of each number,
then use a calculator to find the value.
Give the answers correct to two decimal
places, where necessary.
a) 95
b) 238
c) 11.29
d) 29.09
e) 102.95
f ) 61.74
4 Estimate the cube of each number, then
use a calculator to find the value. Give
the answers correct to two decimal places,
where necessary.
a) 13
b) 48
c) __29
d) 19.11
e) 32.76
f) 136.42
5 Find the value of each expression. Give the
answers correct to two
___ decimal places.
____
7
a) √ 114
b) __
15
_____
√______
c) √ 41.9
d) √68.85
3
_______
3
3
______
e) √ 139.27
f) √ 21.78
6 The figure shows a square with an area
of 315.96 cm2.
Area =
315.96 cm2
a) Calculate the length of one side of
the square. Round your answer to two
decimal places.
b) Calculate the perimeter of the square.
Round your answer to two decimal
places.
7 The figure shows a cube with sides of
4.26 cm.
side = 4.26 cm
a) Calculate the volume of the cube.
Round your answer to two decimal
places.
b) Calculate the area of one face of the
cube. Round your answer to two
decimal places.
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