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Summary: The rules for combining powers and roots seem to confuse a lot of students. They try to memorize everything, and of course it s a big mishmash in their minds. But the laws just come down to counting. which anyone can do, plus three definitions to memorize. This page sorts out what you have to memorize and what you can do based on counting, to solve every problem involving exponents.

See also: Combining Operations (Distributive Laws) includes lots of common mistakes students make, with plenty of exercises to test yourself.

Copying: You re welcome to print copies of this page for your own use, and to link from your own Web pages to this page. But please don t make any electronic copies and publish them on your Web page or elsewhere.

There s nothing mysterious! An exponent is simply shorthand for multiplying that number of identical factors. So 4 is the same as (4)(4)(4), three identical factors of 4. And x is just three factors of x, (x)(x)(x).

One warning: Remember the order of operations. Exponents are the first operation (in the absence of grouping symbols like parentheses), so the exponent applies only to what it s directly attached to. 3x is 3(x)(x)(x), not (3x)(3x)(3x). If we wanted (3x)(3x)(3x), we d need to use grouping: (3x) .

A negative exponent means to divide by that number of factors instead of multiplying. So 4 3 is the same as 1/(4 3 ), and x -3 = 1/x 3 .

As you know, you can t divide by zero. So there s a restriction that x n = 1/x n only when x is not zero. When x = 0, x n is undefined.

A little later, we ll look at negative exponents in the bottom of a fraction .

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.

You can t use counting techniques on an expression like 6 0.1687 or 4.3 . Instead, these expressions are evaluated using logarithms .

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.

1. Write 11 as a multiplication.

2. Write j -7 as a fraction, using only positive exponents.

3. What s the value of 100 ?

4. Evaluate 5 -2 and ( 5) -2 .

Suppose you have (x 5 )(x 6 ); how do you simplify that? Just remember that you re counting factors.

x 5 = (x)(x)(x)(x)(x) and x 6 = (x)(x)(x)(x)(x)(x)

Now multiply them together:

(x 5 )(x 6 ) = (x)(x)(x)(x)(x)(x)(x)(x)(x)(x)(x) = x 11

Why x 11. Well, how many x s are there? Five x factors from x 5. and six x factors from x 6. 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 :

Caution! The rule above works only when multiplying powers of the same base. For instance,

(x 3 )(y 4 ) = (x)(x)(x)(y)(y)(y)(y)

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:

(x )(y ) = (x)(x)(x)(y)(y)(y)

But you know that it doesn t matter what order you do your multiplications in or how you group them. Therefore,

(x)(x)(x)(y)(y)(y) = (x)(y)(x)(y)(x)(y) = (xy)(xy)(xy)

But from the very definition of powers, you know that s the same as (xy) . 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) you can decompose it to (4 )(x ), and if you see (4 )(x ) you can combine it as (4x) .

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/x n comes into play here.

Example: What is x 8 x 6. Well, there are several ways to work it out. One way is to say that x 8 x 6 = x 8 (1/x 6 ), but using the definition of negative exponents that s just x 8 (x -6 ). Now use the product rule (two powers of the same base) to rewrite it as x 8+(-6). or x 8-6. or x 2. Another method is simply to go back to the definition: x 8 x 6 = (xxxxxxxx) (xxxxxx) = (xx)(xxxxxx) (xxxxxx) = (xx)(xxxxxx xxxxxx) = (xx)(1) = x 2. 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. x y can t be combined because it s just xxx/yy; But x y is xxx/yyy, which is (x/y)(x/y)(x/y), which is (x/y) .

What about dividing by a negative power, like y 5 /x 4. Use the rule you already know for dividing:

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/(x 4 ), and 1/(x 4 ) = x 4 .

Write each of these as a single positive power. (I ve slipped in one or two that can t be simplified, just to keep you on your toes.)

5. a 7 b 7

6. 11 2

7. 8 x

8. 5 4 5 6

What do you do with an expression like (x 5 ) 4. There s no need to guess work it out by counting.

(x 5 ) 4 = (x 5 )(x 5 )(x 5 )(x 5 )

Write this as an array:

x 5 = (x) (x) (x) (x) (x)

x 5 = (x) (x) (x) (x) (x)

x 5 = (x) (x) (x) (x) (x)

x 5 = (x) (x) (x) (x) (x)

How many factors of x are there? You see that there are 5 factors in each row from x 5 and 4 rows from ( ) 4. in all 5 4=20 factors. Therefore,

(x 5 ) 4 = x 20

As you might expect, this applies to any power of a power: you multiply the exponents. For instance, (k -3 ) -2 = k (-3)(-2) = k 6. In general,

I can just hear you asking, So when do I add exponents and when do I multiply exponents? Don t try to remember a rule work it out! When you have a power of a power. you ll always have a rectangular array of factors, like the example above. Remember the old rule of length width, so the combined exponent is formed by multiplying. On the other hand, when you re only multiplying two powers together. like g 2 g 3. that s just the same as stringing factors together,

g 2 g 3 = (gg)(ggg) = (ggggg) = g 5

You can always refresh your memory by counting simple cases, like

x 2 x 3 = (xx)(xxx) = x 5

versus

(x 2 ) 3 = (xx) 3 = (xx)(xx)(xx) = x 6

11. (x 4 ) -5

12. (5x )

You probably know that anything to the 0 power is 1. But now you can see why. Consider x 0 .

By the division rule. you know that x 3 /x 3 = x (3 3) = x 0. But anything divided by itself is 1, so x 3 /x 3 = 1. Things that are equal to the same thing are equal to each other: if x 3 /x 3 is equal to both 1 and x 0. then 1 must equal x 0. Symbolically,