Section 6-1

Rules of Exponents

This chapter reviews the properties of exponents. If you want a fast reference, all the properties are listed in a table at the end of this page.

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PRODUCT OF POWERS PROPERTY

How do you simplify 7² × 7⁶?

 

If you recall the way exponents are defined, you know that this means:
(7 × 7) × (7 × 7 × 7 × 7 ×7 × 7)

If we remove the parentheses, we have the product of eight 7s, which can be written more simply as:
7⁸

This suggests a shortcut: all we need to do is add the exponents!
7² · 7⁶ = 7^{(2 + 6)} = 7⁸

In general, for all real numbers a, b, and c,

 

a^b · a^c = a^{(b + c)}

If you remember only this one and forget the rest, you can use it to figure out most of the other properties.

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ZERO EXPONENTS

Many beginning students think it's weird that anything raised to the power of zero is 1. ("It should be 0!") You can use the product of powers property to show why this must be true.

7⁰ · 7¹ = 7^{(0 + 1)} = 7¹

We know 7¹ = 7. So, this says that 7⁰ · 7 = 7. What number times 7 equals 7? If we try 0, we have 0 · 7 = 7. No good.

In general, for all real numbers a, a ≠ 0, we have:
a⁰ = 1

Note that 0⁰ is undefined. (Click here to see why.)

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NEGATIVE EXPONENTS

You can use the product of powers property to figure this one out also. Suppose you want to know what 5^–2 is.

5^–2 × 5² = 5^{(–2 + 2)} = 5⁰

We know 5² = 25, and we know 5⁰ = 1. So, this says that 5^–2 × 25 = 1. What number times 25 equals 1? That would be its multiplicative inverse, 1/25.

In general, for all real numbers a and b, where a ≠ 0, we have:

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QUOTIENT OF POWERS PROPERTY

When you multiply two powers with the same base, you add the exponents. So when you divide two powers with the same base, you subtract the exponents. In other words, for all real numbers a, b, and c, where a ≠ 0,

What you're really doing here is cancelling common factors from the numerator and denominator. Example:

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POWER OF A PRODUCT PROPERTY

When you multiply two powers with the same exponent, but different bases, things go a little differently.

3² · 4² = (3 · 3) · (4 · 4)

Because of the commutative and associative properties of multiplication, we can rewrite this as

3² · 4² = (3 · 4) · (3 · 4) = 12²

In general, for all real numbers a, b, and c (as long as both a and c or both b and c are not zero):
a^c · b^c = (ab)^c

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POWER OF A QUOTIENT PROPERTY

This is pretty similar to the last one. By canceling common factors, you can see that:

For all real numbers a, b, and c (as long as b ≠ 0, and a and c are not both 0):

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RATIONAL EXPONENTS

We've covered positive exponents, negative exponents, and zero exponents. But what if you have an exponent which is not an integer? What, for instance, is 9^1/2?

We can fall back again on the product of powers property to find out:

9^1/2 × 9^1/2 = 9^{(1/2 + 1/2)} = 9¹

We know 9¹ = 9. So we need a number which, when multiplied by itself, gives 9. There are two answers... 3 and –3.

9^1/2 = ±3

So, exponents of 1/2 work a lot like square roots. Similarly, a^1/3 is the same as

and in general

There is an important difference here: the square root of 9 is usually defined as positive 3, whereas 9^1/2 means either 3 or –3 (in most books, at least).

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RECAP

Zero Exponent Property	a0 = 1, (a ≠ 0)
Negative Exponent Property
Product of Powers Property	ab × ac = a(b + c)   (a ≠ 0)
Quotient of Powers Property
Power of a Product Property	ac × bc = (ab)c (a, b ≠ 0)
Quotient of a Product Property
Rational Exponent Property