It’s the Law — the Laws of Exponents

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.

What Is an Exponent, Anyway?

There’s nothing mysterious! An exponent is simply shorthand for multiplying that number of identical factors. So 4³ is the same as (4)(4)(4), three identical factors of 4. And x³ 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. 3x³ is 3(x)(x)(x), not (3x)(3x)(3x). If we wanted (3x)(3x)(3x), we’d need to use grouping: (3x)³.

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.

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

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.

Arbitrary Exponents

You can’t use counting techniques on an expression like 6^0.1687 or 4.3^π. Instead, these expressions are evaluated using logarithms.

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.

Now You Try It!

1. Write 11³ as a multiplication.

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

3. What’s the value of 100^½?

4. Evaluate −5⁻² and (−5)⁻².

[ Answers ]

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 eleven 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)(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:

(x³)(y³) = (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)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.

As that example illustrates, you can combine like exponents even when the bases are different:

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.

Now You Try It!

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. a⁷ ÷ b⁷

6. 11² × 2³

7. 8³ x³

8. 5⁴ × 5⁶

9. p¹¹ ÷ p⁶

10. r⁻¹¹ ÷ r⁻²

[ Answers ]

Powers of Powers

What do you do with an expression like (x⁵)⁴? There’s no need to guess—you can 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 factors. Therefore,

(x⁵)⁴ = x²⁰

As you might expect, this applies to any power of a power: you multiply the exponents. For instance, (k⁻³)⁻² = k^(−3)(−2) = k⁶. 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 g²g³, that’s just the same as stringing factors together,

g²g³ = (gg)(ggg = (ggggg) = g⁵

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

x²x³ = (xx)(xxx) = x⁵

versus

(x²)³ = (xx)³ = (xx)(xx)(xx) = x⁶

Now You Try It!

Perform the operations to remove parentheses:

11. (x⁴)⁻⁵

12. (5x²)³

[ Answers ]

The Zero Exponent

You probably know that anything to the 0 power is 1. But now you can see why. Consider x⁰.

By the division rule, you know that x³/x³ = x⁽³⁻³⁾ = x⁰. But anything divided by itself is 1, so x³/x³ = 1. Things that are equal to the same thing are equal to each other: if x³/x³ is equal to both 1 and x⁰, then 1 must equal x⁰. Symbolically,

x⁰ = x⁽³⁻³⁾ = x³/x³ = 1

There’s one restriction. You saw that we had to create a fraction to figure out x⁰. But division by 0 is not allowed, so our evaluation works for anything to the 0 power except zero itself:

Evaluating 0⁰ is a topic for your calculus course.

Now You Try It!

What is the value of each of these?

13. (a⁶b⁸c¹⁰ / a⁵b⁶d⁷)⁰

14. 17x⁰

[ Answers ]

Radicals

The laws of radicals are traditionally taught separately from the laws of exponents, and frankly I’ve never understood why. A radical is simply a fractional exponent: the square (2nd) root of x is just x^1/2, the cube (3rd) root is just x^1/3, and so on. With this fact at your disposal, you’re in good shape.

Example: . That’s easy to evaluate! You know that the square root of x is x^1/2 and the cube root of that is (x^1/2)^1/3. Then use the power-of-a-power rule to evaluate that as x^(1/2)(1/3) = x^(1/6), which is the 6th root of x.

Example: . Why? Because the square root is the 1/2 power, and the product rule for the same power of different bases tells you that (x^1/2)(y^1/2) = (xy)^1/2.

Fractional or Rational Exponents

So far we’ve looked at fractional exponents only where the top number was 1. How do you interpret x^2/3, for instance? Can you see how to use the power rule? Since 2/3 = (2)(1/3), you can rewrite x^2/3 = x^(2)(1/3) = (x²)^1/3, which is . It works the other way, too: 2/3 = (1/3)(2), so x^2/3 = x^(1/3)(2) = (x^1/3)² = . These are examples of the general rule:

When a power and a root are involved, the top part of the fractional exponent is the power and the bottom part is the root.

Suppose p and r are the same? Then you have, for instance, . But that’s the same as x^5/5, and 5/5=1, so it’s the same as x¹ or just x.

Now You Try It!

15. Write √x⁵ as a single power.

16. Simplify ³√(a⁶b⁹)  (That’s the cube root or third root of a⁶b⁹.)

17. Find the numerical value of 27^4/3 without using a calculator.

[ Answers ]

Conclusion

Well, there you are: the laws of exponents and radicals demystified! Just remember the three basic definitions. When you’re not sure about a rule, like the product rule, don’t try to remember it, just work it out by counting and you’ll do just fine.

Answers

From What Is an Exponent, Anyway? — 1. 11×11×11   2. 1/j⁷   3. √100 = 10   4. −5⁻² = −1/25 and (−5)⁻² = 1/25 (Excel returns 1/25 or 0.04 for both of these, but that’s wrong.

From Multiplying and Dividing Powers — 5. (a/b)⁷   6. cannot be simplified as a power expression. Numerically it’s 121×8 = 968.   7. (8x)³   8. 5¹⁰ (not 25¹⁰!)   9. p⁵   10. r^{−11−(−2)} = r⁻⁹ = 1/r⁹

From Powers of Powers — 11. x⁻²⁰ or 1/x²⁰   12. Use the rule for powers of different bases to start with: 5³(x²)³. Then apply the power-of-a-power rule to get 5³x⁶ or 125x⁶

From The Zero Exponent — 13. 1, provided that a, b, and d are all nonzero   14. 17×1 = 17, provided that x≠0

From Radicals — 15. x^5/2   16. (a⁶b⁹)^1/3 = a²b³   17. 27^4/3 = (27^1/3)⁴ by the power-of-a-power law. 27^1/3 is the same as the cube root of 27, which is 3. (27^1/3)⁴ = 3⁴ = 81