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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

**Squares**, or square numbers, are found when we multiply an integer (whole number) by itself.

E.g.

4 × 4 can be written as ^{2}*“4 squared”* or *“4 to the power of 2”*.

For GCSE Mathematics you need to know squares up to ^{2}

\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 ^{2}

We can arrange ^{2}

We can arrange ^{2}

We can arrange ^{2}

E.t.c

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.

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

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}

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.*

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

DOWNLOAD FREEGet your free square numbers worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE**Squares and square roots** is part of our series of lessons to support revision on **powers and roots**. You may find it helpful to start with the main powers and roots lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

**Term**

A single number (constant) or variable.

E.g.

In the expression

**Coefficients**

The number which the variable is being multiplied by.

E.g.

In ^{3}

**Integer**

A whole number.

E.g.

**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 ^{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 ^{2 }^{2}

**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.

**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.

What is

So,

What is

So,

What is the positive square root of

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

The question only needs the positive answer.

So,

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

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

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

Martin says ^{2} = 6

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

The question focus is on squaring

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

The mistake Martin made was he did ^{2}’

Daphne says the square root of a number is always smaller than the original number.

Prove Daphne is incorrect.

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

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

Perform the operation

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

Clearly state the answer within the context of the question

Daphne is incorrect because:

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

Therefore the positive square root of

The sum of two square numbers is

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

Remember sum means add.

Therefore you are looking for two square numbers that add together to make

It will help here to list the square numbers up to

Perform the operation

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

Clearly state the answer within the context of the question

The two square numbers are

A square has an area of ^{2}. What is the length of one side?

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

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

Therefore you need to find the square root of

Perform the operation

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

Clearly state the answer within the context of the question

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

You must include the units so the final answer is

The length of one side of a square is

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

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

Perform the operation

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

Clearly state the answer within the context of the question

You must include the units so the final answer is ^{2}

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

Is Rebecca right? Explain your answer.

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

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

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

Perform the operation

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:

Clearly state the answer within the context of the question

Rebecca is wrong because

**Square numbers**

Incorrect understanding of squaring a number.

E.g. ^{2} = 9

Not knowing the square numbers up to and including

**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

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

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)**

3. Find the value of:

(a) \sqrt{25}

(b) \sqrt{144}

(c) \sqrt{400}

**(3 Marks)**

Show answer

(a) 5

**(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)**

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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