Aside: Exponents Made Easy

To learn about exponents, we will use the example, 3², which is read in one of two ways: three squared, or 3 to the power of 2.

Three is the base number (written on the base line) and two is the exponent (the little superscript number).

Even in this simple little example, there are some hidden assumptions:

  • Positive numbers are not preceded by a +, or PLUS sign, (even though negative values are preceded by a -, or MINUS sign).  Unless a base number or exponent has a negative sign in front of it, we can assume it is positive.
  • We do not generally use the exponent ¹, because it does not change the value of the base number (number on the base line).  3¹ = 3.  On the other hand, an exponent of -1 would change the value, so it must be used when applicable.  3¯¹ = 1/3.

It has served me well to think that there is an implied +1 multiplied by any number I see standing alone.  You will soon see why this little trick is useful and recommended.

Here is how I would think of 3²:  +1 (+3  x  +3).  An exponent of two tells us to multiply a positive one by the base number, in this case +3, two times.

If we see the number 3 all by itself, it means 3¹.  Multiply positive one (implied) by positive three (the base number) only one time (says the exponent of 1).  1 x 3 = 3.

If we see the number 3°, it means multiply +1 by 3 ZERO times.  In other words, any number with a zero exponent means you do nothing to the implied +1.  This is why it helps to imagine a positive one before all base numbers.  3° = 1.

If you see a negative exponent, it is telling us to DIVIDE our initial +1 by the base number given, however many times the exponent indicates.

3¯² means +1 divided by (+3  x  +3), or +1/9.  By convention, we would drop the + sign and state our answer as 1/9.

Summary: the exponent tells you how many times you multiply (if exponent is positive) or divide (if exponent is negative) the +1 by the base number.

Okay, let’s return now, to find the values for our variables, ‘a’ and ‘b’.