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# Negative Exponents and Scientific Notation

## Overview

• Section 5.2 in the textbook
– Expressions with negative exponents
– Simplifying more complex exponential
expressions
–Writing numbers in scientific notation
– Scientific notation to standard form

## Expressions with Negative Exponents

• Consider x2 / x6
x-4 by the quotient rule
x·x / x·x·x·x·x·x
1 / x4
• Usually, we do NOT leave an expression
with a negative exponent
• Flipping an exponent AND its base from
the numerator into the denominator (or
vice versa) reverses the sign of the
exponent

– The sign of the exponent DOES NOT affect
the sign of the base!
–Whenever using the quotient rule, the initial
result goes into the numerator

## Expressions with Negative Exponents (Example)

Ex 1: Simplify – leave NO negative exponents:

## Reviewing the Exponent Rules

• Product Rule:
• Power Rule:
– Power of a Product:
– Power of a Quotient:
• Quotient Rule:

## Simplifying More Complex Exponential Expressions

• When solving more complicated
expressions:
– Simplify inside of the parentheses using the
product and quotient rules if possible
– Apply the power (if one exists)
• Use the power rule with the exponents
• Evaluate the bases as with normal numbers
– If necessary, write the final answer with
positive exponents

## Simplifying More Complex Exponential Expressions (Example)

Ex 2: Simplify – leave NO negative
exponents:

Ex 3: Simplify – leave NO negative
exponents:

Ex 4: Simplify – leave NO negative
exponents:

## Writing Numbers in Scientific Notation

• Scientific Notation: any number in the
form of a x 10b where -10 < a < 10, a ≠ 0
and b is an integer
– Used to write extreme numbers (large or
small) in a compact format
• To write a number in scientific notation:
– One non-zero number to the left of the
decimal point – the rest to the right

– Determine where to place the decimal point:
• Count how many places the decimal point is
moved
• If the original number (without the sign) is greater
than 1, b (the exponent) is positive
• If the original number (without the sign) is less than
1, b is negative

## Writing Numbers in Scientific Notation (Example)

Ex 5: Write each in scientific notation:
a) -238.41
b) 0.00584
c) 0.018

## Scientific Notation to Standard Form

• Standard Notation: writing a number in
scientific notation without the power of ten
– Take the decimal and move it:
• To the right if b (the exponent) is positive
• To the left if b (the exponent) is negative
• Fill in empty spots with zeros

## Scientific Notation to Standard Form (Example)

Ex 6: Write in standard notation:

## Multiplying or Dividing in Scientific Notation

• Multiply or divide the numbers as normal
• Use the Product or Quotient Rules to
simplify the power of tens
• Write the final answer in scientific notation

## Multiplying or Dividing in Scientific Notation (Example)

Ex 7: Simplify and write as a single number
in scientific notation:

## Summary

• After studying these slides, you should know
how to do the following:
– Simplify expressions containing negative exponents
– Apply the exponent rules to simplify more complex
exponential expressions
– Writing numbers in scientific notation
– Converting scientific notation to standard form
– Multiply or divide using scientific notation