What are the Laws of Exponents?
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Negative Exponents
A negative exponent means to divide by that number of factors instead of multiplying. So 4−3 is the same as 1/(43), and x-3 = 1/x3.
As you know, you can’t divide by zero. So there’s a restriction that x−n = 1/xn only when x is not zero. When x = 0, x−n is undefined.
Fractional Exponents
A fractional exponent — specifically, an exponent of the form 1/n — means to take the nth root instead of multiplying or dividing. For example, 4(1/3) is the 3rd root (cube root) of 4.
Here’s All You Need to Memorize
And that’s it for memory work. Period. If you memorize these three definitions, you can work everything else out by combining them and by counting:
Granted, there’s a little bit of hand waving in my statement that you can work everything else out. Let me make good on that promise, by showing you how all the other laws of exponents come from just the three definitions above. The idea is that you won’t need to memorize the other laws — or if you do choose to memorize them, you’ll know why they work and you’ll find them easier to memorize accurately.
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Multiplying and Dividing Powers
Two Powers of the Same Base
Suppose you have (x5)(x6); how do you simplify that? Just remember that you’re counting factors.
x5 = (x)(x)(x)(x)(x) and x6 = (x)(x)(x)(x)(x)(x)
Now multiply them together:
(x5)(x6) = (x)(x)(x)(x)(x)(x)(x)(x)(x)(x)(x) = x11
Why x11? Well, how many x’s are there? Five x factors from x5, and six x factors from x6, makes 11 x factors total. Can you see that whenever you multiply any two powers of the same base, you end up with a number of factors equal to the total of the two powers? In other words, when the bases are the same, you find the new power by just adding the exponents:
Powers of Different Bases
Caution! The rule above works only when multiplying powers of the same base. For instance,
(x3)(y4) = (x)(x)(x)
If you write out the powers, you see there’s no way you can combine them.
Except in one case: If the bases are different but the exponents are the same, then you can combine them. Example:
(x³)(y³) = (x)(x)(x)
But you know that it doesn’t matter what order you do your multiplications in or how you group them. Therefore,
(x)(x)(x) = (x)(x)(x) = (xy)(xy)(xy)
But from the very definition of powers, you know that’s the same as (xy)³. And it works for any common power of two different bases:
It should go without saying, but I’ll say it anyway: all the laws of exponents work in both directions. If you see (4x)³ you can decompose it to (4³)(x³), and if you see (4³)(x³) you can combine it as (4x)³.
Dividing Powers
What about dividing? Remember that dividing is just multiplying by 1-over-something. So all the laws of division are really just laws of multiplication. The extra definition of x-n as 1/xn comes into play here.
Example: What is x8÷x6? Well, there are several ways to work it out. One way is to say that x8÷x6 = x8(1/x6), but using the definition of negative exponents that’s just x8(x-6). Now use the product rule (two powers of the same base) to rewrite it as x8+(-6), or x8-6, or x2. Another method is simply to go back to the definition: x8÷x6 = (xxxxxxxx)÷(xxxxxx) = (xx)(xxxxxx)÷(xxxxxx) = (xx)(xxxxxx÷xxxxxx) = (xx)(1) = x2. However you slice it, you come to the same answer: for division with like bases you subtract exponents, just as for multiplication of like bases you add exponents:
But there’s no need to memorize a special rule for division: you can always work it out from the other rules or by counting.
In the same way, dividing different bases can’t be simplified unless the exponents are equal. x³÷y² can’t be combined because it’s just xxx/yy; But x³÷y³ is xxx/yyy, which is (x/y)(x/y)(x/y), which is (x/y)³.
Negative Powers on the Bottom
What about dividing by a negative power, like y5/x−4? Use the rule you already know for dividing:
5 5 5 4 5 4
y y y x y x 4 5
--- = -------- = -------- * -- = ----- = x y
-4 ( 4) ( 4) 4
x (1 / x ) (1 / x ) x 1 But that’s much too elaborate. Since 1 / (1/x) is just x, a negative exponent just moves its power to the other side of the fraction bar. So x−4 = 1/(x4), and 1/(x−4) = x4.
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Powers of Powers
What do you do with an expression like (x5)4? There’s no need to guess — work it out by counting.
(x5)4 = (x5)(x5)(x5)(x5)
Write this as an array:
x5 = (x) (x) (x) (x) (x)
x5 = (x) (x) (x) (x) (x)
x5 = (x) (x) (x) (x) (x)
x5 = (x) (x) (x) (x) (x)
How many factors of x are there? You see that there are 5 factors in each row from x5 and 4 rows from ( )4, in all 5×4=20 facto
A negative exponent means to divide by that number of factors instead of multiplying. So 4−3 is the same as 1/(43), and x-3 = 1/x3.
As you know, you can’t divide by zero. So there’s a restriction that x−n = 1/xn only when x is not zero. When x = 0, x−n is undefined.
Fractional Exponents
A fractional exponent — specifically, an exponent of the form 1/n — means to take the nth root instead of multiplying or dividing. For example, 4(1/3) is the 3rd root (cube root) of 4.
Here’s All You Need to Memorize
And that’s it for memory work. Period. If you memorize these three definitions, you can work everything else out by combining them and by counting:
Granted, there’s a little bit of hand waving in my statement that you can work everything else out. Let me make good on that promise, by showing you how all the other laws of exponents come from just the three definitions above. The idea is that you won’t need to memorize the other laws — or if you do choose to memorize them, you’ll know why they work and you’ll find them easier to memorize accurately.
--------------------------------------------------------------------------------
Multiplying and Dividing Powers
Two Powers of the Same Base
Suppose you have (x5)(x6); how do you simplify that? Just remember that you’re counting factors.
x5 = (x)(x)(x)(x)(x) and x6 = (x)(x)(x)(x)(x)(x)
Now multiply them together:
(x5)(x6) = (x)(x)(x)(x)(x)(x)(x)(x)(x)(x)(x) = x11
Why x11? Well, how many x’s are there? Five x factors from x5, and six x factors from x6, makes 11 x factors total. Can you see that whenever you multiply any two powers of the same base, you end up with a number of factors equal to the total of the two powers? In other words, when the bases are the same, you find the new power by just adding the exponents:
Powers of Different Bases
Caution! The rule above works only when multiplying powers of the same base. For instance,
(x3)(y4) = (x)(x)(x)
If you write out the powers, you see there’s no way you can combine them.
Except in one case: If the bases are different but the exponents are the same, then you can combine them. Example:
(x³)(y³) = (x)(x)(x)
But you know that it doesn’t matter what order you do your multiplications in or how you group them. Therefore,
(x)(x)(x)
= (x)(x)(x) = (xy)(xy)(xy)But from the very definition of powers, you know that’s the same as (xy)³. And it works for any common power of two different bases:
It should go without saying, but I’ll say it anyway: all the laws of exponents work in both directions. If you see (4x)³ you can decompose it to (4³)(x³), and if you see (4³)(x³) you can combine it as (4x)³.
Dividing Powers
What about dividing? Remember that dividing is just multiplying by 1-over-something. So all the laws of division are really just laws of multiplication. The extra definition of x-n as 1/xn comes into play here.
Example: What is x8÷x6? Well, there are several ways to work it out. One way is to say that x8÷x6 = x8(1/x6), but using the definition of negative exponents that’s just x8(x-6). Now use the product rule (two powers of the same base) to rewrite it as x8+(-6), or x8-6, or x2. Another method is simply to go back to the definition: x8÷x6 = (xxxxxxxx)÷(xxxxxx) = (xx)(xxxxxx)÷(xxxxxx) = (xx)(xxxxxx÷xxxxxx) = (xx)(1) = x2. However you slice it, you come to the same answer: for division with like bases you subtract exponents, just as for multiplication of like bases you add exponents:
But there’s no need to memorize a special rule for division: you can always work it out from the other rules or by counting.
In the same way, dividing different bases can’t be simplified unless the exponents are equal. x³÷y² can’t be combined because it’s just xxx/yy; But x³÷y³ is xxx/yyy, which is (x/y)(x/y)(x/y), which is (x/y)³.
Negative Powers on the Bottom
What about dividing by a negative power, like y5/x−4? Use the rule you already know for dividing:
5 5 5 4 5 4
y y y x y x 4 5
--- = -------- = -------- * -- = ----- = x y
-4 ( 4) ( 4) 4
x (1 / x ) (1 / x ) x 1 But that’s much too elaborate. Since 1 / (1/x) is just x, a negative exponent just moves its power to the other side of the fraction bar. So x−4 = 1/(x4), and 1/(x−4) = x4.
--------------------------------------------------------------------------------
Powers of Powers
What do you do with an expression like (x5)4? There’s no need to guess — work it out by counting.
(x5)4 = (x5)(x5)(x5)(x5)
Write this as an array:
x5 = (x) (x) (x) (x) (x)
x5 = (x) (x) (x) (x) (x)
x5 = (x) (x) (x) (x) (x)
x5 = (x) (x) (x) (x) (x)
How many factors of x are there? You see that there are 5 factors in each row from x5 and 4 rows from ( )4, in all 5×4=20 facto