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The Decimal System and the Numbers up to 100
What Is the Real Number Line?
How to Compare Numbers by Size
Pairs of Ten and Groups of Ten up to 100
Activity Set 1
What Are Even and Odd Numbers?
What Is the Place-value System?
Activity Set 1
Sets of Numbers
Interval and Set Notation
What Are Number Sequences?
Examples of Mathematical Sequences
How Do You Find the n-th Term in a Sequence?
How Do Arithmetic Sequences Work?
Square Numbers, Cube Numbers and Triangular Numbers
Recursive and Explicit Formulas for Sequences
What Are Series in Math?
What Are Arithmetic Series?
How to Calculate Amortizing Loans with Series Formulas
Geometric Series Formulas
Infinite Geometric Series and Convergence
How Is Fixed-rate Mortgage Calculated?
How to Calculate Present Values Using Timelines
How to Calculate Future Values Using Timelines
Examples of Calculations with Series
What Is Decimal Notation?
Which Number Is in the Tenths Place?
Which Number Is in the Hundredths Place?
What Does Negative Numbers Mean?
How to Calculate with Negative Numbers
Prime Numbers
Activity Set 1
Prime Factorization Using the Division Method
Prime Factorization Using a Factor Tree
How to Write Scientific Notation
How to Calculate with Scientific Notation

Here, you will take a look at how you do calculations with numbers in scientific notations. It’s a piece of cake. The main idea is to work separately with the power of 10’s and the numbers in front of the powers.

Rule

Multiplication with Scientific Notations

(a ⋅ 10n) ⋅(b ⋅ 10m) = a ⋅ 10n ⋅ b ⋅ 10m = a ⋅ b ⋅ 10n+m

(a ⋅ 10n) ⋅(b ⋅ 10m) = a ⋅ 10n ⋅ b ⋅ 10m = a ⋅ b ⋅ 10n+m

Rule

Division with Scientific Notation

a ⋅ 10n b ⋅ 10m = a b ⋅ 10n 10m = a b ⋅ 10n−m

Example 1

Calculate (4 ⋅ 104) ⋅(2 ⋅ 102)

You use the multiplication rule.

(4 ⋅ 104) ⋅(2 ⋅ 102) = 4 ⋅ 2 ⋅ 104+2 = 8 ⋅ 106

(4 ⋅ 104) ⋅(2 ⋅ 102) = 4 ⋅ 2 ⋅ 104+2 = 8 ⋅ 106

Example 2

Calculate 5 ⋅ 104 ⋅ 3 ⋅ 105

Remember to sort.

5 ⋅ 104 ⋅ 3 ⋅ 105 = 5 ⋅ 3 ⋅ 104+5 = 15 ⋅ 109 = 1.5 ⋅ 10 ⋅ 109 = 1.5 ⋅ 109+1 = 1.5 ⋅ 1010

A value in scientific notation has a number from 1 to 10 multiplied by a power of 10. Since 5 ⋅ 3 =15, you will need to convert 15 to a number between 1 and 10. To do this, write 15 = 1.5 ⋅ 10 (in scientific notations) and repeat once more.

Example 3

Calculate 8 ⋅ 106 4 ⋅ 104

You use the division rule.

8 ⋅ 106 4 ⋅ 104 = 8 4 ⋅106 104 = 2 ⋅ 106−4 = 2 ⋅ 102 = 200

8 ⋅ 106 4 ⋅ 104 = 8 4 ⋅106 104 = 2 ⋅ 106−4 = 2 ⋅ 102 = 200

Example 4

Calculate 3 ⋅ 109 6 ⋅ 105 and write in scientific notation

You factorize into two fractions

3 ⋅ 109 6 ⋅ 105 = 3 6 ⋅109 105 = 0.5 ⋅ 109−5 = 0.5 ⋅ 104 = 5 ⋅ 10−1 ⋅ 104 = 5 ⋅ 103

A value in scientific notation has a number from 1 to 10 multiplied by a power of 10. You must convert 0.5 to a number between 1 and 10. To do this, write 0.5 = 5 ⋅ 10−1 (in scientific notation) and repeat once more.

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How to Convert Scientific Notation to Standard Notation
What Are Complex Numbers?
What Does the Complex Plane Mean?
What Is Polar Form for Complex Numbers?
What Is the Norm and the Argument of a Complex Number?
What Is Euler’s Formula for Complex Numbers?
What Are Complex Conjugates?
Addition and Subtraction of Complex Numbers
Activity Set 1
How to Multiply Complex Numbers
How Can You Simplify Complex Fractions
What Are the Roots of a Complex Number?
What Is de Moivre's Formula and How Do You Use It?
How to Find the Distance Between Two Complex Numbers
How to Find the Midpoint of Two Complex Numbers
How to Solve Linear Equations with Complex Numbers
How to Solve Quadratic Equations with Complex Numbers
How to Factorize Complex Polynomials
What Is the Fundamental Theorem of Algebra?
Addition with Help from the Number Line
Subtraction with Help from the Number Line
Addition and Subtraction (The Place-value System)
How to Add Using the Column Method
How to Subtract Using the Column Method
Multiplication as Repeated Addition
The Times Tables
Which Products Give the Answers in the Times Tables?
How to Multiply by 10
How to Multiply by 100
How to Multiply by 1000
How Long Multiplication Works
What Is Doubling in Math?
What Does Halving Mean in Math?
Division as Inverse Multiplication
What Are the Divisibility Rules?
How to Divide by 10
How to Divide by 100
How to Divide by 1000
How to Do Long Division
Examples of Long Division
What Are Powers in Math?
What Are the Power Rules?
What Are Powers of 10?
How Square Roots Work
Square Roots of Products and Fractions
Order of Operations (MDAS)
Parentheses and Order of Operations
Order of Operations (PEMDAS)
Signs in Math (Positive or Negative)
How to Do Mental Math (Rounding)
How to Do Mental Math (Estimates)
How to Round a Number
How to Make Estimates (Mental Math)
How to Calculate the Cost per Unit
What Are Fractions?
What Is the Value of a Fraction?
Proper Fraction, Improper Fraction and Mixed Fraction
How Do You Expand Fractions?
How Do You Simplify Fractions?
How to Simplify a Fraction by Canceling
Add and Subtract Fractions with Equal Denominators
Add and Subtract Fractions with Different Denominators
Find the Common Denominator by Multiplying Denominators
Find the Common Denominator by Expanding or Simplifying
Find the Common Denominator Using the Times Tables
How to Find the Common Denominator Using Factorization
How to Multiply Fractions
How to Multiply Fractions by Cross-canceling
How to Divide Fractions
What Does Percentage Mean?
How Can You Calculate Percentages?
How to Find Relative Percent Change
How to Calculate Growth Factor
How to Use Growth Factors in Calculations
How Do Percentage Points Work?
Convert Percentages to Fractions and Decimal Numbers
Convert Fractions and Decimal Numbers to Percentages
How to Convert Between Decimal Numbers and Fractions
Converting Fractions, Decimal Numbers and Percentages
What Are the Units of Measurement in the Metric System?
What Are the Metric Units for Length?
What Are the Metric Units for Area?
What Are the Metric Units for Volume?
What Are the Metric Units for Weight?
What Are the Prefixes for SI Units?
How to Find Volumetric Mass Density (Metric)
What Is Temperature and What Are Degrees Celsius?
How Do You Calculate Metric Units of Measurement?
Choosing Appropriate Metric Units of Measurement
What Are the Units of Measurement in the Imperial System?
Activity Set 1
What Are the Imperial Units for Length?
What Are the Imperial Units for Area?
Activity Set 1
What Are the Imperial Units for Volume?
Activity Set 1
What Are the Imperial Units for Weight?
Activity Set 1
How to Find Volumetric Mass Density (Imperial)
What Is Temperature and What Are Degrees Fahrenheit?
How Do You Calculate Imperial Units of Measurement?
How Do You Convert Imperial to Metric?
How Do You Convert Metric to Imperial?
The Digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)
How Do Significant Figures Work?
What Does Measurement Uncertainty Mean?
What Does Relative Measurement Uncertainty Mean?
What Is Digital and Analog Clock?
Learning About the Clock (Seconds)
Learning About the Clock (Minutes)
Activity Set 1
Learning About the Clock (Hours)
Activity Set 1
Learning About the Clock (Whole and Half Hours)
Learning About the Clock (Quarters of an Hour)
Learning About the Clock (Ten To and Ten Past)
Learning About the Clock (Five To and Five Past)
Mental Arithmetic with Time
Calculations with the Clock
What Is Day and Night?
What Are Days and Weeks?
What Are Months?
What Are the Seasons of the Year?
How Does Scale Ratio Work?
How to Read Blueprint Scale
What Is the Ratio Between Two Numbers?
What Does Mixing Ratio Mean?
How to Find Distance if You Have Velocity and Time
What Is a Speed Diagram?
Coins of the United States Dollar
Banknotes of the United States Dollar
How Do You Count Money?
How Math Is Used in Shopping
How Currency Exchange Works
How Can Personal Finance Best Be Managed?
What Are the Different Types of Income?
How Does Commission Salary Work?
How Does Tax and Tax Basis Work?
How Do Bonus Payments Work?
How Value-added Tax Is Calculated
How Credit Card Interest Rates Work
Consumer Price Index and Purchasing Power of a Dollar
What Are Price Index and Consumer Price Index?
What Are Real Wages in Economics?
How Do Loans Work?
What Is an Amortizing Loan?
What Are Fixed-rate Mortgages?
How to Budget Your Money
How to Use Spreadsheets for Accounting
Interest Over One Period
Interest Over Several Periods
How to Compute Compound Interest on Savings
What Do Revenue, Cost, Unit Cost and Profit Mean?
How Do Price and Demand Affect Revenue?
Marginal Revenue, Marginal Cost and Marginal Profit
How to Find Cost-minimizing Input
How to Find Profit-maximizing Input
What Is a Vector?
What Does Vector Coordinates Mean?
What is a Position Vector?
Activity Set 1
Sum and Difference of Vectors
Activity Set 1
Numbers multiplied by a vector
Activity Set 1
How to Find the Vector Between Two Points
What Is the Length of a Vector?
Finding the Distance Between Two Points Using Vectors
How to Find Parallel Vectors
How to Find the Scalar Product of Two Vectors (2D)
How Can You Find the Angle Between Two Vectors?
What Does Unit Vector Mean?
What Are Basis Vectors Used For?
Find the Medians of a Triangle Using Vector Calculation
How to Parametrize a Line
How to Solve Vector Equations
What Are Vector Functions?
Vectors in Three-dimensional Spaces
How to Find the Scalar Product of Two Vectors (3D)
What Is the Vector Product?
How Does the Right-hand Rule for Cross Products Work?
How to Calculate Area of a Parallelogram with Vectors
How to Calculate Area of a Triangle with Vectors
Calculate Volume of Prism and Pyramid with Vectors
What Does Parameterization Mean?
Parameterization of Lines in Space
Parameterization of a Plane
How to Determine the Angle Between Two Lines
How Do You Find the Angle Between a Line and a Plane?
How to Find the Angle Between Two Planes
How to Find the Distance from a Point to a Line
How to Find the Distance Between Two Lines
How to Find the Distance Between a Point and a Plane
How to Find the Distance Between a Line and a Plane
How to Find the Distance Between Two Planes
What Is the Equation for a Sphere?
How to Find the Surface Area of a Spherical Segment
The Intersection Between Planes and Coordinate Axes
How to Find the Intersection Between Line and Plane
How to Find the Intersection Between Two Planes
Intersection Between a Tangent Plane and a Sphere
The Intersection Between a Plane and a Sphere
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