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Chapter 12. Exponents and Polynomials. Negative Exponents and Scientific Notation. 12.2. Negative Exponents. Using the quotient rule,. But what does x -2 mean?. Negative Exponents.

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negative exponents
Negative Exponents

Using the quotient rule,

But what does x-2 mean?

negative exponents1
Negative Exponents

In order to extend the quotient rule to cases where the difference of the exponents would give us a negative number we define negative exponents as follows.

If a is a real number other than 0, and n is an integer, then


Helpful Hint

Simplify by writing each result using positive exponents only.

Don’t forget that since there are no parentheses, x is the base for the exponent –4.


Simplify by write each result using positive exponents only.




Simplify by writing each of the following expressions with positive exponents.

(Note that to convert a power with a negative exponent to one with a positive exponent, you simply switch the power from the numerator to the denominator, or vice versa, and switch the exponent to its opposite value.)

summary of exponent rules
Summary of Exponent Rules

Power of a quotient

Quotient rule for exponents

Negative exponent

Product rule for exponentsam·an = am+n

If m and n are integers and a and b are real numbers, then:

Power rule for exponents (am)n = amn

Power of a product (ab)n = an· bn

Zero exponent a0 = 1, a≠ 0



Simplify by writing the following expression with positive exponents.


Scientific Notation

In many fields of science we encounter very large or very small numbers. Scientific notation is a convenient shorthand for expressing these types of numbers.

A positive number is written in scientificnotation if it is written as the product of a number a, where 1 ≤ a < 10, and an integer power r of 10: a×10r.


Scientific Notation

To Write a Number in Scientific Notation

Step 1: Move the decimal point in the original number so that the new number has a value between 1 and 10.

Step 2: Count the number of decimal places the decimal point is moved in Step 1. If the original number is 10 or greater, the count is positive. If the original number is less than 1, the count is negative.

Step 3: Multiply the new number in Step 1 by 10 raised to an exponent equal to the count found in Step 2.




Move the decimal 3 places to the left, so that the new number has a value between 1 and 10.

Move the decimal 4 places to the right, so that the new number has a value between 1 and 10.




Write each of the following in scientific notation.

Since we moved the decimal 3 places, and the original number was > 10, our count is positive 3.

4700 = 4.7 103

Since we moved the decimal 4 places, and the original number was < 1, our count is negative 4.

0.00047 = 4.7 10-4


Scientific Notation

In general, to write a scientific notation number in standard form,move the decimal point the same number of spaces as the exponent on 10. If the exponent is positive, move the decimal point to the right. If the exponent is negative, move the decimal point to the left.



5.2738 103


6.45 10-5


Write each of the following in standard notation.

Since the exponent is a positive 3, we move the decimal 3 places to the right.

5.2738 103

= 5273.8

Since the exponent is a negative 5, we move the decimal 5 places to the left.

00006.45 10-5

= 0.0000645



(7.3 10-2)(8.1 105)


Operations with Scientific Notation

Multiplying and dividing with numbers written in scientific notation involves using properties of exponents.


Perform the following operations.

= (7.3 · 8.1) (10-2· 105)

= 59.13 103

= 59,130