The magnitude of an earthquake represents the exponent *m* in the expression

Valdivia, Chile has suffered two major earthquakes. The 1575 Valdivia earthquake had a magnitude of 8.5. The world's largest earthquake was the 1960 Valdivia earthquake at a magnitude of 9.5.

What was the size of the 1575 earthquake compared to the 1960 one?

Source: Wikipedia

### Negative and Zero Exponents

In this concept, we will introduce negative and zero exponents.

#### Zero Exponents

Step 1: Evaluate

Step 2: What is a number divided by itself? Apply this to Step 1.

Step 3: Fill in the blanks.

#### Negative Exponents

Step 1: Expand

Step 2: Evaluate

Step 3: Are the answers from Step 1 and Step 2 equal? Write them as a single statement.

Step 4: Fill in the blanks.

From the the discussions above, we have learned two very important properties of exponents. First, anything to the zero power is one. Second, negative exponents indicate placement. If an exponent is negative, it needs to be moved from where it is to the numerator or denominator.

Let's simplify the following expressions. Your answer should only have positive exponents.

5255

x7yz12x12yz7

a4b0a8b

or

xy58y−3

27g−7h018g

Now, let's multiply the following two fractions together and simplify. The answer should only have positive exponents.

The easiest way to approach this problem is to multiply the two fractions together first and then simplify.

### Examples

#### Example 1

Earlier, you were asked to find the size of the 1575 earthquake compared to the 1960 earthquake.

Set each earthquake's magnitude up as an exponential expression and divide.

Therefore, the size of the 1575 earthquake was

**Simplify the following expressions.**

#### Example 2

#### Example 3

#### Example 4

2a8b−416a−5⋅43a−3b0a4b7=128a8−3b−416a−5+4b7=8a5+1b7+4=8a6b11

### Review

Simplify the following expressions. Answers cannot have negative exponents.

8284 x6x15 7−37−2 y−9y10 x0y5xy7 a−1b8a5b7 14c10d−421c6d−3 8g0h30g−9h2 5x410y−2⋅y7xx−1y g9h56gh12⋅18h3g8 4a10b712a−6⋅9a−5b420a11b−8 −g8h6g−8⋅9g15h9−h11 - Rewrite the following exponential pattern with positive exponents:
5−4,5−3,5−2,5−1,50,51,52,53,54 . - Evaluate each term in the pattern from #13.
- Fill in the blanks.

As the numbers increase, you ______________ the previous term by 5.

As the numbers decrease, you _____________ the previous term by 5.

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 6.2.