In the study of indices, symbols are written with two sizes and in two different positions. They have their own unique meanings.
a2 means a times a, or simply a x a.
(a + b)2 means (a + b) x (a + b).
(anything)3 means (anything) x (anything) x (anything).
Therefore, from above examples, we can see that "anything" operated by a small number higher up above it means repetition by that number of times (defined by that little number).
Mistakes normally made:
(a + b)2 ==> a2 + b2 This is wrong!
a2 - b2 ==> (a - b) 2 This is also incorrect!
The "2" is a power and not a factor. It means repetition. And therefore cannot be factorised.
The "2" is written above the normal line (called the base), and thus has "bigger" power than the base element.
The correct answer to the mistakes:
(a + b)2 = ( a + b) ( a + b) = a2 + 2ab + b2
a2 - b2 = ( a x a ) - ( b x b ) = (a - b ) ( a + b ).
To summarise, the "smaller" number (or letter) makes the "larger" base repeats the number of times indicated by that small number.
Principles of mathematics and its indices' concept .....
Understand the principles and concepts, and you will be fine.
For more common mistakes in indices, click here.