GCSE Maths Number

Powers and Roots

Square Numbers and Square Roots

# Squares And Square Roots

Here we will learn about squares 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 square root of an integer. You’ll also learn how to solve problems by applying knowledge of squares.

There are also squares 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?

Squares, or square numbers, are found when we multiply an integer (whole number) 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 squares up to 152 :

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

## Perfect squares

The square number of an integer is also called a perfect square.

We can square numbers with decimal places but we do not refer to these as square numbers or perfect squares.

A given number or variable that is ‘squared’ is multiplied by itself and this applies to decimals as to integers.

## 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}

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. For example the square root of 225 is 15

A square root is the inverse operation of squaring a number.
The square root symbol looks like this:

$\sqrt{ \; \; \; }$

Its mathematical name is the ‘radical’.  The square root function only gives us the positive value of the square root. We call this the principal square root.

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

This means when we want the 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 \pm \sqrt{9}=\pm 3 \\\\ \text { As } 8^{2}=64 \text { ,the square root of } 64 \text { is } 8 \text { or } −8 & \therefore \pm \sqrt{64}=\pm 8 \end{array}

In GCSE mathematics, if asked to find the square root of a number you would only need to give the positive value of a square root which is the principal square root.

## 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.

## Squares 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 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 squares 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

## Squares and square root problems examples

### Example 4: knowledge of squares

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 a number is always smaller than the original number.
Prove Daphne is incorrect.

You are looking for a relationship between the square root of a number and the original number. It doesn’t have to be an integer, it could be a fraction or decimal. You could try finding the square root of 0.25.

\begin{array}{l} \sqrt{0.25}=0.5 \end{array}

Daphne is incorrect because:

\begin{array}{l} \sqrt{0.25}=0.5 \end{array}

Therefore the positive square root of 0.25 is 0.5 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.

$\pm \sqrt{81}=\pm 9$

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”.

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. When solving an equation of the form x^{2} = 100 You need to find the postive and negative square roots of 100.

x=\pm \sqrt{100} = \pm10 not 10

### Practice square numbers and square roots questions

1. What is 4 squared?

2

44

8

16

4 squared =4 multiplied by 4 = 16

2. What is the positive square root of 100?

10

10000

1

50

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

3. What is 7^2?

14

77

49

70

7^{2}=7 × 7=49

4.  What is x if x^{2} = 144 ?

12 and -12

12 and

11 and 12

11 and -11

12 × 12=144 and -12 × -12=144

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

1 and 24

4 and 25

4 and 16

9 and 16

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

112.5cm

56.2cm

15cm

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)

(a) 3 \times 3

9

(1)

(b) 3 \times 4

12

(1)

(c) 36 \times 4

144

(1)

3.  Find the value of:

(a) \sqrt{25}

(b) \sqrt{144}

(c) \sqrt{400}

(3 Marks)

(a)

(1)

(b) 12

(1)

(c) 20

(1)

4. Simplify the following expression:

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

(2 Marks)

{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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