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Powers and Roots

Square Numbers and Square Roots

Square Numbers and Square Roots

Here we will learn about square numbers and square roots including what a square number is and what a square root is, as well as how to square a number and how to find the square root of an integer. You’ll also learn how to solve problems by applying knowledge of square numbers.

There are also square numbers and square roots worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck

What is a square number?

A square number is found when we multiply an integer (whole number) by itself, these are sometimes called ‘perfect squares’.
We can square numbers with decimals places but we do not refer to these as square numbers.

A number/variable that is ‘squared’ is multiplied by itself.

E.g.
4 × 4 can be written as 42 and is spoken as “4 squared” or “4 to the power of 2”.

For GCSE Mathematics you need to know the square numbers up to 152 :

\begin{align*} 1 \times 1&=1 \quad \quad \quad \quad \quad \quad \quad \quad \quad \text { Therefore } 1 \text { is a square number } \\\\ 2 \times 2&=4 \quad \quad \quad \quad \quad \quad \quad \quad \quad \text { Therefore } 4 \text { is a square number } \\\\ 3 \times 3&=9 \quad \quad \quad \quad \quad \quad \quad \quad \quad \text { Therefore } 9 \text { is a square number } \\\\ 4 \times 4&=16 \; \; \quad \quad \quad \quad \quad \quad \quad \quad \text { Therefore } 16 \text { is a square number } \\\\ 5 \times 5&=25 \; \; \quad \quad \quad \quad \quad \quad \quad \quad \text { Therefore } 25 \text { is a square number } \\\\ 6 \times 6&=36 \; \; \quad \quad \quad \quad \quad \quad \quad \quad \text { Therefore } 36 \text { is a square number } \\\\ 7 \times 7&=49 \; \; \quad \quad \quad \quad \quad \quad \quad \quad \text { Therefore } 49 \text { is a square number } \\\\ 8 \times 8&=64 \; \; \quad \quad \quad \quad \quad \quad \quad \quad \text { Therefore } 64 \text { is a square number } \\\\ 9 \times 9&=81 \; \; \quad \quad \quad \quad \quad \quad \quad \quad \text { Therefore } 81 \text { is a square number } \\\\ 10 \times 10&=100 \quad \quad \quad \quad \quad \quad \quad \quad \text { Therefore } 100 \text { is a square number } \\\\ 11 \times 11&=121 \quad \quad \quad \quad \quad \quad \quad \quad \text { Therefore } 121 \text { is a square number } \\\\ 12 \times 12&=144 \quad \quad \quad \quad \quad \quad \quad \quad \text { Therefore } 144 \text { is a square number } \\\\ 13 \times 13&=169 \quad \quad \quad \quad \quad \quad \quad \quad \text { Therefore } 169 \text { is a square number } \\\\ 14 \times 14&=196 \quad \quad \quad \quad \quad \quad \quad \quad \text { Therefore } 196 \text { is a square number } \\\\ 15 \times 15&=225 \quad \quad \quad \quad \quad \quad \quad \quad \text { Therefore } 225 \text { is a square number } \end{align*}

A square number can be represented as an array which forms the shape of a square. 

E.g.

We can arrange 12 as a square which has side lengths of 1 unit. 

We can arrange 22 as a square which has side lengths of 2 units.

We can arrange 32 as a square with side lengths of 3 units. 

We can arrange 42 as a square which has side lengths of 4 units.

E.t.c

What is a square number?

What is a square number?

Squaring negative numbers

We can also square negative numbers.

E.g.

\begin{array}{ll} (-5) \times(-5)=25 & \therefore(-5)^{2}=25 \\\\ (-7) \times(-7)=49 & \therefore(-7)^{2}=49 \end{array}

You will notice that when we square a negative number we get a positive number.
This is because a negative number multiplied by a negative number gives us a positive result.
Learn more by reviewing our lesson on negative numbers. 

When we square negative 5 we get the same answer as when we square positive 5.
This is true for all numbers (and variables)

E.g.

\begin{array}{l} 4^{2} \text { and }(-4)^{2} \text { both equal } 16\\\\ 10^{2} \text { and }(-10)^{2} \text { both equal } 100\\\\ x^{2} \text { and }(-x)^{2} \text { both equal } x^{2}\\\\ \end{array}

What is a square root?

The square root of a number is a value that can be multiplied by itself to give the original number.
A square root is the inverse operation of squaring a number.
The square root function looks like this:

\[\sqrt{ \; \; \; } \]

Its mathematical name is the ‘radical’.  

When we square a positive number/variable or a negative number/variable we always get a positive answer.


This means when we square root a number/variable we get a positive and a negative answer.

The symbol ± means both the positive and negative of the number/variable shown. 

E.g.

\begin{array}{l} \text { As } 3^{2}=9 \text { ,the square root of } 9 \text { is } 3 \text { or } −3 & \therefore \sqrt{9}=\pm 3 \\\\ \text { As } 8^{2}=64 \text { ,the square root of } 64 \text { is } 8 \text { or } −8 & \therefore \sqrt{64}=\pm 8 \end{array}

We call the positive value of a square root the principal square root.

What is a square root?

What is a square root?

Square numbers and square roots worksheet

Square numbers and square roots worksheet

Square numbers and square roots worksheet

Get your free square numbers and square roots worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Square numbers and square roots worksheet

Square numbers and square roots worksheet

Square numbers and square roots worksheet

Get your free square numbers and square roots worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE

Key words

Term

A single number (constant) or variable.

E.g.
In the expression 4x − 7 both 4x and − 7 are terms


Coefficients

The number which the variable is being multiplied by.

E.g.
In 2x3 the coefficient is 2


Integer

A whole number.

E.g.
1, 7 or 1003


Index (also called exponent or powers)

The index number is the number of times you multiply a number/variable by itself.

E.g.
The index number in 52 is 2

Note: the plural of index is indices.

Note: you will see index number as a superscript.


Base number

The number/variable that is being multiplied by itself an amount of times.

E.g.
The base number in 52 is 5 and in 2x2 the base number is x

Advanced vocabulary –  only for Additional Maths, A-Level

Real numbers

Any positive or negative number is called a real number. Numbers that are not ‘real’ are called imaginary numbers. Integers, decimals, fractions are all examples of real numbers.

E.g.
1, 2, 5 and 100 are examples of a positive real numbers.

-1, -2, -50 and -65.67 are examples of a negative real numbers.


Imaginary numbers

Numbers that are not real are called imaginary numbers, for example you will notice we cannot find a real square root of a negative number (try it on a calculator), this because it is an imaginary number. Numbers that contain an imaginary part and real part are called complex numbers.

Square numbers and square root examples

Example 1

What is 5 squared?

5 squared means 5 × 5

So, 5 squared is 25.

Example 2

What is 15 squared?

15 squared means 15 × 15

So, 15 squared is 225.

Example 3

What is the positive value of the square root of 64?

The square root of 64 means what value can be multiplied by itself to give 64.

The question only needs the positive answer.

So,

\[\sqrt{64}=8.\]

How to use square numbers and square roots

In order to solve problems involving square numbers and square roots :

  1. Identify whether you need to square or square root the number/variable
  2. Perform the operation
  3. Clearly state the answer within the context of the question e.g. including units, only the positive value

Square numbers and square root problems examples

Example 4: knowledge of square numbers

Martin says 32 = 6. Why is Martin wrong? What mistake did he make?

  1. Identify whether you need to square or square root the number/variable

The question focus is on squaring 3 or “3 squared”.

2 Perform the operation

\begin{array}{l} 3 \times 3 = 3^{2} = 9 \end{array}

3 Clearly state the answer within the context of the question

Martin is wrong because 3 squared is 9.

The mistake Martin made was he did ‘3 × 2’ not ‘32.

Example 5: problem solving with square roots

Daphne says the square root of an integer is always smaller than the original number.
Prove Daphne is incorrect.

You are looking for a relationship between the square root of an integer and the original number. It will help to write down your square numbers and their square roots.

\begin{array}{l} \sqrt{100}=\pm 10 \\\\ \sqrt{81}=\pm 9 \\\\ \sqrt{64}=\pm 8 \\\\ \sqrt{49}=\pm 7 \\\\ \sqrt{36}=\pm 6 \\\\ \sqrt{25}=\pm 5 \\\\ \sqrt{16}=\pm 4 \\\\ \sqrt{9}=\pm 3 \\\\ \sqrt{4}=\pm 2 \\\\ \sqrt{1}=\pm 1 \end{array}

Daphne is incorrect because:

\[\sqrt{1}=\pm 1\]


Therefore the square root of 1 could be 1 or -1 which is not smaller than the original number.

Example 6: solving problems involving square numbers

The sum of two square numbers is 61. Find the two square numbers.

Remember sum means add.
Therefore you are looking for two square numbers that add together to make 61.


It will help here to list the square numbers up to 61.

1, 4 , 9 , 16 , 2536 , 49


You now need to pick two of these numbers that when added together make 61.

The two square numbers are 25 and 36.

Example 7: square numbers within a 2D polygon

A square has an area of 81cm2. What is the length of one side?

The area of a square is found by multiplying the length and width together.
For a square the length and width are the same length.


Therefore you are looking for a number that when multiplied by itself (or squared) is equal to 81.


Therefore you need to find the square root of 81.

\[\sqrt{81}=\pm 9\]

The length of a side cannot be negative therefore the answer is 9.


You must include the units so the final answer is 9cm.

Example 8: square numbers within a 2D polygon

The length of one side of a square is 5cm. What is the area of the square?

The area of a square is found by multiplying the length and width together. For a square the length and width are the same length.


Therefore we are going to multiply 5 by itself or “square” it.

\begin{array}{l} 5 \times 5 =5^{2} =25 \end{array}

You must include the units so the final answer is 25cm2

Example 9

Rebecca says “when you add three consecutive square numbers, the answer is always odd”.

Is Rebecca right? Explain your answer.

To prove Rebecca wrong we only need to find one example where she is incorrect, this is sometimes known as proof by contradiction.


Therefore we are looking for 3 consecutive square numbers that when added together give an odd answer.


It will be helpful here to list the ‘main’ square numbers.

The square numbers:

\[1, 4 , 9 , 16 , 25, 36 , 49, 64, 81, 100 , 121 , 144 , 169, 196, 225 \]


You are now looking for one example where three of these numbers when added together make an odd number.


For example: 1 + 4 + 9 = 14

Rebecca is wrong because 1 + 4 + 9 = 14

Common misconceptions

  • Square numbers

Incorrect understanding of squaring a number.

E.g. 32 = 9 NOT 6

Not knowing the square numbers up to and including 15.

  • Square roots

Only recognising positive values of square roots of integers.

E.g. \sqrt{100} = \pm10 not 10

Practice square numbers and square roots questions

1. What is 4 squared?

2
GCSE Quiz False

44
GCSE Quiz False

8
GCSE Quiz False

16
GCSE Quiz True

4 squared =4 multiplied by 4 = 16

2. What is the positive value of the square root of 100?

10
GCSE Quiz True

10000
GCSE Quiz False

1
GCSE Quiz False

50
GCSE Quiz False

10 \times 10 = 100 therefore \sqrt{100}=10

3. What is 7^2?

14
GCSE Quiz False

77
GCSE Quiz False

49
GCSE Quiz True

70
GCSE Quiz False

7^{2}=7 × 7=49

4.  What is \sqrt{144} ?

12 and -12

GCSE Quiz True

12 and

GCSE Quiz False

11 and 12

GCSE Quiz False

11 and -11

GCSE Quiz False

12 × 12=144 and -12 × -12=144

5.  The sum of two square numbers is 29 . Find the two square numbers.

1 and 24

GCSE Quiz False

4 and 25

GCSE Quiz True

4 and 16

GCSE Quiz False

9 and 16

GCSE Quiz False

The square numbers up to 29 are 1, 4, 9, 16 and 25.

 

The two square numbers that add up to 29 are 4 and 25.

6.  A square has an area of 225 cm^2. What is the length of one side?

22.5cm

GCSE Quiz False

112.5cm

GCSE Quiz False

56.2cm

GCSE Quiz False

15cm

GCSE Quiz True

To find the area of a square, we multiply the side lengths together.

 

Since the height and width of a square are equal, we need to find a number which, when multiplied by itself, makes 225 . Therefore we need to find the positive square root of 225.

 

\sqrt{225}=15

Square numbers and square roots GCSE questions

1.  Work out the value of:

 

(a) 3^{2} 

 

(b) 3 \times 2^{2} 

 

(c) 6^{2} \times 2^{2} 

 

(3 Marks)

Show answer

(a) 3 \times 3 

 

9

(1)

 

(b) 3 \times 4 

 

12

(1)

 

(c) 36 \times 4 

 

144

(1)

2. Here is a list of numbers:

 

100 \quad \quad 18 \quad \quad 6 \quad \quad 25 \quad \quad 7 \quad \quad 144 \quad \quad 1 \quad \quad 19 

 

(a) List the square numbers.

 

(b) Which number when square rooted is \pm10

 

(2 Marks)

Show answer

(a) 25, 100 , 144 and 1

(1)

 

(b) 100

(1)

3.  Find the positive value of:

 

(a) \sqrt{25} 

 

(b) \sqrt{144}

 

(c) \sqrt{400}

 

(3 Marks)

Show answer

(a)

(1)

 

(b) 12 

(1)

 

(c) 20 

(1)

4. Simplify the following expression:

 

4^{2} \times \sqrt{x^{2}}

 

(2 Marks)

Show answer
{16x}

 

Correct coefficient

(1)

 

Correct x (or x^{1} )

(1)

Learning checklist

You have now learned how to:

  • Calculate square numbers up to 15 × 15
  • Use positive integer powers and their associated real roots
  • Recognise and use the square numbers
  • Apply properties of squares to a context

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