This topic could actually be included in my series "A Parent's Guide To Algebra's Basic Concepts" and people who are following that series might want to print this and place it with those articles.
In that series, we have not yet reached the point of needing the laws of exponents.
I am writing this article as an answer to a question.
It IS a good question.
First, I want to point out that the word "equation" in the title is not really important.
The idea or concept of the Laws of Exponents is simplifying expressions or individual terms involving exponents whether those expressions are in equations or not.
There is just one set of Laws of Exponents, not one for expressions and a different set for equations.
That should make you feel better already! Algebra students often have trouble with both remembering the laws and differentiating them (knowing which one to use).
This really is a very serious problem because if the rules get switched in your head, you will make the same mistake several times which means it gets "practiced" and learned.
Once a mistake gets learned, it is very difficult to correct.
It is much better to never make the mistake in the first place.
Hopefully, this article will clear up any confusion you may have.
Before we look at the Laws, let's review what exponents are and why they exist.
We also need to review some names so we are all thinking the same thing at the same time.
A typical number example involving the use of an exponent would look like: 3^4 and is read as three to the fourth power.
The 3 is called the base of the term or just "the base.
" The 4 is called the exponent; and the entire term 3^4 is called "a power.
" It means (3)(3)(3)(3) or 3 (the base) multiplied by itself 4 (the exponent) times.
In symbols, an example might look like x^2.
This would be read as x to the second power or x squared and would mean (xx).
The x is the base, the 2 is the exponent, and x^2 is called a power.
To remove some confusion about exponents, we are going to look at two "similar" situations that students tend to treat as the same.
However, they are very different.
Only one of these uses exponents.
First: x + x + x + x is read as x plus x plus x plus x.
The second: (x)(x)(x)(x) is read as x times x times x times x.
The first situation: x + x + x + x is an example of repeated or repetitive addition.
The other name for repetitive addition is MULTIPLICATION.
Multiplication was created as a short cut or simplified way to write repeated addition.
So, the addition problem of x + x + x + x can be written using multiplication as 4x which means you have x ADDED to itself 4 times.
Using numbers, this would look like: 2 + 2 + 2 + 2 = (2)(4) or (4)(2) = 8.
Remember: repetitive addition = multiplication and no exponents are involved.
The second situation: (x)(x)(x)(x) is NOT repetitive addition.
Instead, we have repeated or repetitive multiplication.
THIS situation is why exponents exist.
Exponents provide the short cut way to simplify repetitive multiplication.
The multiplication expression (x)(x)(x)(x) can be written using an exponent as x^4 which means you have x MULTIPLIED by itself 4 times.
Caution! Big words coming: repetitive multiplication = exponentiation.
In simpler terms, exponents provide the short cut or simplified way to write repeated multiplication.
It is much simpler to write y^6 than (y)(y)(y)(y)(y)(y).
The situations that require either addition or multiplication of exponents will look like either of these: (x^2)(x)(x^3) or (x^2)^4.
The first is read as x squared times x times x cubed.
The second is read as x to the 2nd power raised to the fourth power.
We are going to give these two situations NAMES based on a description of what is actually happening.
First: (x^2)(x)(x^3) we will call "multiplying like bases.
" The other: (x^2)^4 is called "raising a power to a power.
" Think back to the labels (base, exponent, and power) we used earlier.
It is imperative that you understand why these examples are given these names.
The best way to tell if you understand this is to explain it to someone else.
Go get Mom and try explaining why these names are what they are.
If you can, that's awesome and we can move on to the rules.
If you find yourself hesitating (which would be very normal), come back and read through this again and practice speaking out loud.
Then give Mom another try.
I'll wait.
Note: the word "like" in multiplying like bases is significant.
The expression (x^2)(y^3), which would be read as x squared times y cubed, would be an example of multiplying DIFFERENT bases.
There is NO WAY to write this any simpler than it already is.
I'm only using parentheses here because the "^" is the only way I can denote an exponent and it would be confusing to look at without the ( ).
With exponents written in the typical fashion, this would be read as x squared y cubed.
We wouldn't even need to say multiplied by because it is "understood" that when variables are written side by side, multiplication is implied.
The reason most students have trouble remembering rules is that they don't take enough time to understand where the rule came from.
(This is when most students "tune out" the teacher.
) So we aren't going to use any rules...
yet.
We are just going to fall back on the basics--what exponents mean.
Looking at (x^2)(x)(x^3) first, this would mean (xx)(x)(xxx).
This is simply repeated multiplication which can be simplified with an exponent.
Since we have x multiplied by itself 6 times, we can write x^6.
Had we just ADDED the exponents in the initial problem (remember that x has an "understood" exponent of 1), 2 + 1 + 3 we would have the correct exponent.
What about (x^2)^4? Again, falling back on the meaning of exponents, (x^2)^4 would mean (xx)(xx)(xx)(xx).
Again, we just have repeated multiplication: x^8.
Looking at the original problem, the short cut would have been to MULTIPLY the exponents.
To summarize: there are 3 specific situations with exponents to consider.
These are the rules you've been waiting for: 1.
To Multiplying Different Bases: No Shortcut Exists! Example: (x^3)(y^2)z (read as: x cubed y squared z) cannot be simplified.
2.
To Multiplying Like Bases: Add The Exponents! Example: (x^2)(x^3)(x^4).
2 + 3 + 4 = 9.
Answer: x^9 3.
To Raise a Power to a Power:Multiply The Exponents! Example: (x^4)^3.
(4)(3) = 12.
Answer: x^12 Thus, the answer to the original question of: When should exponents be added?When multiplying like bases.