6.1 NORMAL PATH squares
Theory
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 split square roots with multiplication:
√500 = √(5 . 100) = √5 . √100 = √5 .10 = 10√5
Examples
Squares
We have already learned how squares (2-D shapes) look like and what special special features they have:
The edges of a square are equally long.
The angles of a square are 90 degrees.
Instead of writing down 5 . 5 = 25 we can write down 52 = 25, this is a shorter way.
52 is the same as 5 . 5
32 is the same as 3 . 3
262 is the same as 26 . 26
we call the ...2 a squared number. 52 is five squared, 32 is three squared and 262 is 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)2 is the same as -5 . -5 = 25
(-3)2 is the same as -3 . -3 = 9
You do square the minus sign
(-26)2 is the same as -26 . -26 = 676
Without brackets
-52 is the same as -5 . 5 = -25
-32 is the same as -3 . 3 = -9
You don't square the minus sign
-262 is 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
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 are no answers of square roots with a negative number.
However... it is possible if you are going to use imaginary numbers and the complex number theory
If you want to know how this works you can find information in the in-depth path.
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.
Exercises
- Exercise 1
Calculate
a) 42 + 3 - 1
b) 62 - 32
c) (7 - 6)2
d) (1/4)2 + (2/3)2
e) 7 - 62 : 3
- Exercise 2
Calculate
a) (-8)2
b) √81
c) √49 + 42 - 1
d) 3 . √121 + 7 . √625