5.5 Negative Exponents and Scientific Notation

Negative Exponents Using the quotient rule, But what does x -2 mean?

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 Negative Exponents

Example 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. Helpful Hint

Simplify by write each result using positive exponents only. Example

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

If m and n are integers and a and b are real numbers, then: Product rule for exponents a m · a n = a m+n Power rule for exponents (a m ) n = a mn Power of a product (ab) n = a n · b n Power of a quotient Quotient rule for exponents Zero exponent a 0 = 1, a ≠ 0 Negative exponent Summary of Exponent Rules

Simplify by writing the following expression with positive exponents. Example

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 scientific notation if it is written as the product of a number a, where 1 ≤ a < 10, and an integer power r of 10: a × 10 r. 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. Scientific Notation

Write each of the following in scientific notation a b. Example

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. Scientific Notation

Write each of the following in standard notation a b. Example

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

Example Perform the following operations. ( )( ) a. b.