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Middle Years

1.05 Square roots and perfect squares


Square Numbers

We can represent repeated addition using multiplication, for example: 


In a similar way, we can represent repeated multiplication using powers.

To indicate that a number has been multiplied by itself we write the number being multiplied, called the base, and then indicate the number of times it is multiplied by itself by writing this number, called the power (also known as the index or exponent), above and to the right of the base. Here is an example: 


This is read as "three to the power of two" or "three raised to the power of $2$2".

For now we will just be looking at numbers raised to the power of two, known as square numbers or perfect squares. In the above example $3^2$32 is often referred to as "three squared".

What does it mean to 'square' a number? Look at the pattern below!

$1^2$12 $2^2$22 $3^2$32 $4^2$42 $5^2$52
1 4 9 16 25

You can see above that each formation of dots forms a square.

$1^2$12 $=$= $1\times1$1×1 $=$= $1$1  
$2^2$22 $=$= $2\times2$2×2 $=$= $4$4  
$3^2$32 $=$= $3\times3$3×3 $=$= $9$9  
$4^2$42 $=$= $4\times4$4×4 $=$= $16$16 and so on.


Use the applet below to explore the first twelve square numbers.

Finding the square root

If we are asked to find the square root of a value, we are being asked, "What number multiplied by itself would give this value?"

You might also see the square root symbol written with a number inside it, for example, $\sqrt{25}$25.

This means find the square root of $25$25.

Worked example

What does the value of $\sqrt{144}$144 equal?

ThinkWe should read $\sqrt{144}$144 as "the square root of $144$144".

This is the number that squares (multiplies with itself) to make $144$144.

We know that $144=12\times12$144=12×12.

Do: The square root of $144$144 is $12$12.


Practice questions


Evaluate $9^2$92


Evaluate $4^2+9^2$42+92


Evaluate $\sqrt{8^2+6^2}$82+62

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