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Rules of Exponents Name
Example:
base: 5 exponent: 2 power: 2nd read: five squared or five to the second power
Exponents are used in many algebra problems, so it's important that you understand the rules for working with exponents. Let's go over each rule in detail, and see some examples.
Zero Rule According to the "zero rule," any nonzero number raised to the power of zero equals 1. Rules of 1 There are two simple "rules of 1" to remember. First, any number raised to the power of "one" equals itself. This makes sense, because the power shows how many times the base is multiplied by itself. If it's only multiplied one time, then it's logical that it equals itself. Secondly, one raised to any power is one. This, too, is logical, because one times one times one, as many times as you multiply it, is always equal to one.
Product Rule The exponent "product rule" tells us that, when multiplying two powers that have the same base, you can add the exponents. In this example, you can see how it works. Adding the exponents is just a short cut!
x^0 = 1 (as long as x ≠ 0 )
x^1 = x
1 m^ = 1
xm^ ⋅ xn^ = xm + n
42 ⋅ 43 = 4 ⋅ 4 ⋅ 4 ⋅ 4 ⋅ 4 = 42 +^3 = 45
Quotient Rule The quotient rule tells us that we can divide two powers with the same base by subtracting the exponents. You can see why this works if you study the example shown.
Negative Exponents The last rule in this lesson tells us that any nonzero number raised to a negative power equals its reciprocal raised to the opposite or positive power.
Note also: any nonzero number raised to a positive power equals its reciprocal raised to the opposite or negative power.
Power Rule The "power rule" tells us that to raise a power to a power, just multiply the exponents. Here you see that 5^2 raised to the 3rd power is equal to 5^6.
[This also makes sense. Note: ]
xm^ ÷ xn^ = x
m
xn^ =^ x
m − n (Note: x ≠ 0.)
45 ÷ 42 = 4
5
42 =^
4 ⋅ 4 =^4
(Note: Giant One shown using marks. Numbers are not eliminated; they are simplified to 1.)
x −^ n^ = (^) x^1 n 4 −^2 = 412 = 161
xm^ = (^) x^1 − m
53 = (^51) − 3
( xm^ ) n^ = xmn ( 52 )^3 = 52 ×^3 = 56
( 52 )^3 = 52 ⋅ 52 ⋅ 52 = ( 5 ⋅ 5) ⋅ ( 5 ⋅ 5) ⋅ ( 5 ⋅ 5) = 5 ⋅ 5 ⋅ 5 ⋅ 5 ⋅ 5 ⋅ 5 = 56
Although we wouldn’t show our work this way, let’t think of each part of the expression separately, simplify it, and then put them all together.
So,
5. Often times a simplified problem is considered one without negative exponents or parentheses.
Example: 15 x
4 y 5 z 3
25 x^9 y^2 z^3
15 25 =^
3 5
x^4
x^9 =^ x
− 5 or 1
x^5
y^5 y^2 =^ y
3 z^3
z^3 =^1
15 x^4 y^5 z^3 25 x^9 y^2 z^3 =^
3 5 ⋅^
1 x^5 ⋅^ y
(^3) ⋅ 1 = 3 y^3 5 x^5
