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Arithmetic Negative numbers

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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 :

\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?

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.

What is a square root?

Square numbers worksheet

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

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Square numbers worksheet

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

DOWNLOAD FREE

Related lessons on powers and roots

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:

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 5²is 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 5²is 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 :

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

Squares and square root problems examples

Example 4: knowledge of squares

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

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.

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 0.25.

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 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.

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 61.

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

Perform the operation

1, 4 , 9 , 16 , 25, 36 , 49

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

Clearly state the answer within the context of the question

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?

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 81.

Therefore you need to find the square root of 81.

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 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?

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 5 by itself or “square” it.

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 25cm²

Example 9

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 3 consecutive square numbers that when added together give an odd answer.

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: 1 + 4 + 9 = 14

Clearly state the answer within the context of the question

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

2

44

8

16

4 squared =4 multiplied by 4 = 16

10

10000

1

50

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

14

77

49

70

$7^{2}=7$ × $7=49$

$12$ and $-12$

$12$ and

$11$ and $12$

$11$ and $-11$

$12$ × $12=144$ and $-12$ × $-12=144$

$1$ 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 $25.$

$22.5$ cm

$112.5$ cm

$56.2$ cm

$15$ cm

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) $3 \times 2^{2}$

(3 Marks)

Show answer

(a) $3 \times 3$

$9$

(1)

(b) $3 \times 4$

$12$

(1)

$144$

(1)

3. Find the value of:

(a) $\sqrt{25}$

(b) $\sqrt{144}$

(3 Marks)

Show answer

(a) $5$

(1)

(b) $12$

(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

The next lessons are

Still stuck?

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