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5.1 Exponents and Scientific Notation

Exponents Exponents that are natural numbers are shorthand notation for repeating factors. 3 4 = is the base 4 is the exponent (also called power) Note by the order of operations that exponents are calculated before other operations.

Product Rule for Exponents If m and n are positive integers and a is a real number, then a m · a n = a m+n The Product Rule

Use the product rule to simplify. 3 2 · 3 4 = 3 6 = 3 · 3 · 3 · 3 · 3 · 3= 729 z 3 · z 2 · z 5 = z (3y 2 )(– 4y 4 ) = 3 · y 2 (–4) · y 4 = 3(–4)(y 2 · y 4 ) = –12y 6 = = z 10 Example

Zero Exponent If a does not equal 0, then a 0 = 1. Simplify each of the following expressions. 5 0 = 1 (xyz 3 ) 0 = x 0 · y 0 · (z 3 ) 0 = 1 · 1 · 1 = 1 –x0–x0 = –(x 0 ) = – 1 Zero Exponent Example:

Evaluate the following. 5 0 = 1 (xyz 3 ) 0 = x 0 · y 0 · (z 3 ) 0 = 1 · 1 · 1 = 1 –x0–x0 = –(x 0 )= – 1 Example

The Quotient Rule Quotient Rule for Exponents If a is a nonzero real number and m and n are integers, then

Example Use the quotient rule to simplify. Group common bases together

Negative Exponents If a is a real number other than 0 and n is a positive integer, then

Example Simplify and write with positive exponents only. Remember that without parentheses, x is the base for the exponent –4, not 2x

Example Simplify and write with positive exponents only.

Example Simplify. Assume that a and t are nonzero integers and that x is not 0.

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 a product of a number a, where 1  a < 10, and an integer power r of 10. a  10 r Scientific Notation Scientific notation

1)Move the decimal point in the original number to the until the new number has a value between 1 and 10. 2)Count the number of decimal places the decimal point was 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. 3)Write the product of the new number in Step 1 by 10 raised to an exponent equal to the count found in Step 2. Writing a Number in Scientific Notation

Write each of the following in scientific notation a. Move the decimal 3 places to the left, so that the new number has a value between 1 and 10. Since we moved the decimal 3 places, and the original number was > 10, our count is positive = b. Move the decimal 4 places to the right, so that the new number has a value between 1 and 10. Since we moved the decimal 4 places, and the original number was < 1, our count is negative = Example

Writing a Scientific Notation Number in Standard Form Move the decimal point the same number of places 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. Since the exponent is a positive 3, we move the decimal 3 places to the right = b. Since the exponent is a negative 5, we move the decimal 5 places to the left = Example