Exponents & Scientific Notation

MATH 102Contemporary Math

S. Rook

Overview

• Section 6.5 in the textbook:– Exponent rules– Scientific notation

Exponent Rules

44

Review of Exponential Notation

• Consider (-3)4 What is its expanded form?• What about -34? • Recall an exponential expression is made up

of a base raised to a powerxa = x · x · x · x · x · x · x · … · x (a times)

– Identifying the base is the key

55

Product Rule

• Consider 22 ∙ 24 How does this expand?• Product Rule: xa ∙ xb = xa+b – When multiplying LIKE BASES (the same variable),

add the exponents– Only applies when the operation is multiplication

66

Power Rule

• Consider (22)4 How does this expand?• Power Rule: (xa)b = xab

– When raising variables to a power, multiply the exponents

– Only applies when the exponent is outside a set of parentheses

77

PRODUCT Rule versus POWER Rule

• Be careful not to confuse:– Product Rule: x4 · x7 (multiplying LIKE bases) – Power Rule: (x4)7 (exponent appears with NO

base)– It is a common mistake to mix up the Product

Rule and the Power Rule!

88

Quotient Rule

• Consider 35 / 32 How does this expand?• Quotient Rule: xa / xb = xa-b – When dividing LIKE BASES (the same variable),

subtract the exponents– Only applies when the operation is division

Exponent Rules (Example)

Ex 1: Use the exponent rules to evaluate:

a) 32 x 34

b) (23)2

c) 59 / 57

1010

Expressions with Negative Exponents

• Consider 22 / 26

2-4 by the quotient rule• 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– e.g. 3-2 = 1 / 32

1111

Expressions with Negative Exponents (Continued)

• How would we evaluate 2-3 ? 2-3 ≠ -8

– The sign of the exponent DOES NOT affect the sign of the base!

– Whenever using the quotient rule, the result goes into the numerator

Exponent Rules (Example)

Ex 2: Use the exponent rules to evaluate:

a) 2-4 x 22

b) (33)-2

c) 6-2 / 6-4

Scientific Notation

1414

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:– Place the decimal point so that one non-zero

number is to the left of the decimal point and the rest of the numbers are to the right

Writing Numbers in Scientific Notation (Continued)

– Determine the effect of moving 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

15

Scientific Notation (Example)

Ex 3: Rewrite in scientific notation:

a) 4,356,000b) 0.008

1717

Scientific Notation to Standard Form

• Standard Notation: writing a number expressed in scientific notation without the power of ten– To convert to standard notation, 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 (Example)

Ex 4: Rewrite in standard notation:

a) 4.5 x 10-7

b) 3.25 x 104

1919

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

Scientific Notation (Example)

Ex 5: Perform the following operations and leave the answer in scientific notation:

a) (1.2 x 10-3)(3 x 105)b) (4.8 x 104) / (1.6 x 10-3)c)

4

63

1052.1

1015.4106368.9

Summary

• After studying these slides, you should know how to do the following:– Apply the exponent rules to any numerical base– Convert from standard notation to scientific notation and

vice versa– Multiply and divide numbers in scientific notation

• Additional Practice:– See problems in Section 6.5

• Next Lesson:– Sequences (Section 6.6)