This demonstrates the first basic exponent rule:. Nothing combines. Now that I know the rule namely, that I can add the powers on the same base , I can start by moving the bases around to get all the same bases next to each other:. Now I want to add the powers on the a 's and the b 's. However, the second a doesn't seem to have a power. What do I add for this term? Anything that has no power on it, in a technical sense, being "raised to the power 1 ".
Anything to the power 1 is just itself, since it's "multiplying one copy" of itself. So the expression above can be rewritten as:. In the following example, there are two powers, with one power being "inside" the other, in a sense. To do the simplification, I can start by thinking in terms of what the exponents mean. The "to the fourth" on the outside means that I'm multiplying four copies of whatever base is inside the parentheses. In this case, the base of the fourth power is x 2.
Multiplying four copies of this base gives me:. Each factor in the above expansion is "multiplying two copies" of the variable. This expands as:. This is a string of eight copies of the variable. This demonstrates the second exponent rule:. Whenever you have an exponent expression that is raised to a power, you can simplify by multiplying the outer power on the inner power:.
If you have a product inside parentheses, and a power on the parentheses, then the power goes on each element inside. For instance:. Warning: This rule does NOT work if you have a sum or difference within the parentheses. Exponents, unlike mulitiplication, do NOT " distribute " over addition. When in doubt, write out the expression according to the definition of the power. For instance, given x — 2 2 , don't try to do this in your head.
Instead, write it out; "squared" means "multiplying two copies of", so:. The mistake of erroneously trying to "distribute" the exponent is most often made when the student is trying to do everything in his head, instead of showing his work. Do things neatly, and you won't be as likely to make this mistake. Now that I know the rule about powers on powers, I can take the 4 through onto each of the factors inside.
I'll need to remember that, with the c , inside the parentheses it's "to the power 1 ". Anything to the power zero is just " 1 " as long as the "anything" it not itself zero. This rule is explained on the next page.
In practice, though, this rule means that some exercises may be a lot easier than they may at first appear:.
Who cares about that stuff inside the square brackets? I sure don't, because the zero power on the outside means that the value of the entire thing is just 1.
If two powers have the same base then we can divide the powers. When we divide powers we subtract their exponents. The rule for the power of a power and the power of a product can be combined into the following rule:.
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