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Zero, Negative, and Fractional Exponents

Evaluate constant and variable terms with negative exponents

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Negative and Zero Exponents

The magnitude of an earthquake represents the exponent m in the expression 10m.

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 5656 by using the Quotient of Powers property.

5656=566=50

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

5656=1

Step 3: Fill in the blanks. amam=amm=a=

a0=1

Negative Exponents

Step 1: Expand 3237 and cancel out the common 3’s and write your answer with positive exponents.

3237=333333333=135

Step 2: Evaluate 3237 by using the Quotient of Powers property.

3237=327=35

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

135=35

Step 4: Fill in the blanks. 1am=a and 1am=a

1am=am and 1am=am

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.

  1. 5255

5255=53=153=1125

  1. x7yz12x12yz7

x7yz12x12yz7=y11z127x127=y0z5x5=z5x5

  1. a4b0a8b

a4b0a8b=a48b01=a4b1=1a4b

or

a4b0a8b=1a84b=1a4b

  1. xy58y3

xy58y3=xy5y38=xy5+38=xy88

  1. 27g7h018g

27g7h018g=32g1g7=32g1+7=32g8

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

4x2y520x85x6y15y9

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

4x2y520x85x6y15y9=20x2+6y5+1300x8y9=x2+68y5+1+915=x4y1515=y1515x4

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.

108.5109.5=101=1101=110

Therefore, the size of the 1575 earthquake was 110 the 1960 one.

Simplify the following expressions.

Example 2

8689

8689=869=183=1512

Example 3

3x10y221x7y4

3x10y221x7y4=x107y2(4)7=x3y67

Example 4

2a8b416a543a3b0a4b7

2a8b416a543a3b0a4b7=128a83b416a5+4b7=8a5+1b7+4=8a6b11

Review

Simplify the following expressions. Answers cannot have negative exponents.

  1. 8284
  2. x6x15
  3. 7372
  4. y9y10
  5. x0y5xy7
  6. a1b8a5b7
  7. 14c10d421c6d3
  8. 8g0h30g9h2
  9. 5x410y2y7xx1y
  10. g9h56gh1218h3g8
  11. 4a10b712a69a5b420a11b8
  12. g8h6g89g15h9h11
  13. Rewrite the following exponential pattern with positive exponents: 54,53,52,51,50,51,52,53,54.
  14. Evaluate each term in the pattern from #13.
  15. 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. 

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Vocabulary

TermDefinition
Negative Exponent Property The negative exponent property states that \frac{1}{a^m} = a^{-m} and \frac{1}{a^{-m}} = a^m for a \neq 0.
Zero Exponent Property The zero exponent property says that for all a \neq 0, a^0 = 1.

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