6.1 SUPPORTING PATH squares

Theory

Squares are 2-D shapes because they are flat. The characteristics of squares are that they have 4 edges which are equally long. They also have 4 right-angles (90 degrees). You can see this in the picture below.

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You can calculate the area of a square by using the following formula:

Area square = length x width

Exercise area square:

Calculate the area of the given square.

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Answer

A squared number is a number multiplied by itself.

An example of a squared number is 5² = 25, this is the same as 5 ^.5 = 25 .

Another example is 3² = 9, this is the same as 3 ^.3 = 9.

A square root is the opposite of a squared number.

An example of this is √25 = 5, because 5² = 5 ^.5 = 25

Another example is √9 = 3, because 3² = 3 ^.3 = 9

The square of a square root (√...)²

( √5 )²= √5 ^.√5 = √25 = 5

( √7 )²= √7 ^.√7 = √49 = 7

You can also simplify square roots using multiplication:

√500 = √(5 ^.100) = √5 ^.√100 = √5 ^.10 = 10√5

√32= √(16 ^.2) = √16^.√2= √2 ^.4 = 4√2

Examples

Squares

You have just seen that when you are calculating the area of a square you need to multiply the length of the edges by itself. In the exercise above you did 8 ^.8 = 64.

A shorter way to write a number multiplied by itself is 8²this is exactly the same as 8 ^. 8.

So, a number multiplied by itself is called a squared number.

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A couple of examples:

Instead of writing down 5 ^. 5 = 25 we can write down 5²= 25, this is a shorter way.

5²is the same as 5 ^. 5

3² is the same as 3^.3

26²is the same as 26 ^.26

we call the ...²a squared number. 5² is five squared, 3² is three squared and 26²is twenty-six squared.

You can also write the square numbers of negative numbers. Pay attention between the difference of negative numbers with and without brackets.

With brackets

(-5)²is the same as -5 ^. -5 = 25

(-3)² is the same as -3^.-3 = 9

You do square the minus sign

(-26)²is the same as -26 ^.-26 = 676

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Without brackets

-5²is the same as -5 ^. 5 = -25

-3² is the same as -3^.3 = -9

You don't square the minus sign

-26²is the same as -26 ^.26 = -676

Square roots

A square root (√) is the opposite of a square.

√25 = 5 because 5 ^. 5 = 5² = 25

√9 = 3 because 3 ^.3 = 3² = 9

√676 = 26 because 26 ^.26 = 26²= 676

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!!You can not take the square root of a negative number because there is not a squared number where the answer has a negative outcome!!

Example:

√-16 = no answer

Because 4 ^. 4 = 4²= 16 --> not -16

-4 ^.-4 = (-4)² = 16 --> not -16

So, the answer will never be -16 and that's why there is no answer (n.a.) of a square root with a negative number.

-√16 = -4

Because the -1 in front of the square root is actually a invisible -1 multiplied by the square root. You will get the following calculation:

= -1 ^.√16

= -1 ^. 4

= -4

The square of a square root √..²

You already know that opposite of squaring a number is taking the square root of a number.

So, √81 = 9 because 9²= 9 ^.9 = 81

But, what happens if we square a square root?

Examples:

( √9 )²= √9 ^.√9 = √81 = 9

( √9 )²= ( 3 )² = 9

( √4 )²= √4 ^.√4 = √16 = 4

( √4 )²= ( 2 )²= 4

( √5 )²= √5 ^.√5 = √25 = 5

( √7 )²= √7 ^.√7 = √49 = 7

You can see that the answer of the square root squared is equal to the number underneath the square root. A square and a square root cancel each other out.