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Scientific Notation

- Calculus
- Absolute Maxima and Minima
- Absolute and Conditional Convergence
- Accumulation Function
- Accumulation Problems
- Algebraic Functions
- Alternating Series
- Antiderivatives
- Application of Derivatives
- Approximating Areas
- Arc Length of a Curve
- Arithmetic Series
- Average Value of a Function
- Calculus of Parametric Curves
- Candidate Test
- Combining Differentiation Rules
- Combining Functions
- Continuity
- Continuity Over an Interval
- Convergence Tests
- Cost and Revenue
- Density and Center of Mass
- Derivative Functions
- Derivative of Exponential Function
- Derivative of Inverse Function
- Derivative of Logarithmic Functions
- Derivative of Trigonometric Functions
- Derivatives
- Derivatives and Continuity
- Derivatives and the Shape of a Graph
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Polar Functions
- Derivatives of Sec, Csc and Cot
- Derivatives of Sin, Cos and Tan
- Determining Volumes by Slicing
- Direction Fields
- Disk Method
- Divergence Test
- Eliminating the Parameter
- Euler's Method
- Evaluating a Definite Integral
- Evaluation Theorem
- Exponential Functions
- Finding Limits
- Finding Limits of Specific Functions
- First Derivative Test
- Function Transformations
- General Solution of Differential Equation
- Geometric Series
- Growth Rate of Functions
- Higher-Order Derivatives
- Hydrostatic Pressure
- Hyperbolic Functions
- Implicit Differentiation Tangent Line
- Implicit Relations
- Improper Integrals
- Indefinite Integral
- Indeterminate Forms
- Initial Value Problem Differential Equations
- Integral Test
- Integrals of Exponential Functions
- Integrals of Motion
- Integrating Even and Odd Functions
- Integration Formula
- Integration Tables
- Integration Using Long Division
- Integration of Logarithmic Functions
- Integration using Inverse Trigonometric Functions
- Intermediate Value Theorem
- Inverse Trigonometric Functions
- Jump Discontinuity
- Lagrange Error Bound
- Limit Laws
- Limit of Vector Valued Function
- Limit of a Sequence
- Limits
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- Limits of a Function
- Linear Approximations and Differentials
- Linear Differential Equation
- Linear Functions
- Logarithmic Differentiation
- Logarithmic Functions
- Logistic Differential Equation
- Maclaurin Series
- Manipulating Functions
- Maxima and Minima
- Maxima and Minima Problems
- Mean Value Theorem for Integrals
- Models for Population Growth
- Motion Along a Line
- Motion in Space
- Natural Logarithmic Function
- Net Change Theorem
- Newton's Method
- Nonhomogeneous Differential Equation
- One-Sided Limits
- Optimization Problems
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- Particle Model Motion
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- Polar Coordinates
- Polar Coordinates Functions
- Polar Curves
- Population Change
- Power Series
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- Riemann Sum
- Rolle's Theorem
- Root Test
- Second Derivative Test
- Separable Equations
- Simpson's Rule
- Solid of Revolution
- Solutions to Differential Equations
- Surface Area of Revolution
- Symmetry of Functions
- Tangent Lines
- Taylor Polynomials
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- Techniques of Integration
- The Fundamental Theorem of Calculus
- The Mean Value Theorem
- The Power Rule
- The Squeeze Theorem
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- Theorems of Continuity
- Trigonometric Substitution
- Vector Valued Function
- Vectors in Calculus
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- Washer Method
- Decision Maths
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- Altitude
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- Area of Plane Figures
- Area of Rectangles
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- Area of Rhombus
- Area of Trapezoid
- Area of a Kite
- Composition
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- Convexity in Polygons
- Coordinate Systems
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- Equation of Circles
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- Figures
- Fundamentals of Geometry
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- Glide Reflections
- HL ASA and AAS
- Identity Map
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- Isometry
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- Law of Cosines
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- Linear Measure and Precision
- Median
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- Perpendicular Bisector
- Plane Geometry
- Polygons
- Projections
- Properties of Chords
- Proportionality Theorems
- Pythagoras Theorem
- Rectangle
- Reflection in Geometry
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- Rotations
- SSS and SAS
- Segment Length
- Similarity
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- Special quadrilaterals
- Squares
- Surface Area of Cone
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- Surface Area of Sphere
- Surface Area of a Solid
- Surface of Pyramids
- Symmetry
- Translations
- Trapezoids
- Triangle Inequalities
- Triangles
- Using Similar Polygons
- Vector Addition
- Vector Product
- Volume of Cone
- Volume of Cylinder
- Volume of Pyramid
- Volume of Solid
- Volume of Sphere
- Volume of prisms
- Mechanics Maths
- Acceleration and Time
- Acceleration and Velocity
- Angular Speed
- Assumptions
- Calculus Kinematics
- Coefficient of Friction
- Connected Particles
- Constant Acceleration
- Constant Acceleration Equations
- Converting Units
- Force as a Vector
- Kinematics
- Newton's First Law
- Newton's Law of Gravitation
- Newton's Second Law
- Newton's Third Law
- Projectiles
- Pulleys
- Resolving Forces
- Statics and Dynamics
- Tension in Strings
- Variable Acceleration
- Probability and Statistics
- Bar Graphs
- Basic Probability
- Charts and Diagrams
- Conditional Probabilities
- Continuous and Discrete Data
- Frequency, Frequency Tables and Levels of Measurement
- Independent Events Probability
- Line Graphs
- Mean Median and Mode
- Mutually Exclusive Probabilities
- Probability Rules
- Probability of Combined Events
- Quartiles and Interquartile Range
- Systematic Listing
- Pure Maths
- ASA Theorem
- Absolute Value Equations and Inequalities
- Addition and Subtraction of Rational Expressions
- Addition, Subtraction, Multiplication and Division
- Algebra
- Algebraic Fractions
- Algebraic Notation
- Algebraic Representation
- Analyzing Graphs of Polynomials
- Angle Measure
- Angles
- Angles in Polygons
- Approximation and Estimation
- Area and Circumference of a Circle
- Area and Perimeter of Quadrilaterals
- Area of Triangles
- Arithmetic Sequences
- Average Rate of Change
- Bijective Functions
- Binomial Expansion
- Binomial Theorem
- Chain Rule
- Circle Theorems
- Circles
- Circles Maths
- Combination of Functions
- Combinatorics
- Common Factors
- Common Multiples
- Completing the Square
- Completing the Squares
- Complex Numbers
- Composite Functions
- Composition of Functions
- Compound Interest
- Compound Units
- Conic Sections
- Construction and Loci
- Converting Metrics
- Convexity and Concavity
- Coordinate Geometry
- Coordinates in Four Quadrants
- Cubic Function Graph
- Cubic Polynomial Graphs
- Data transformations
- Deductive Reasoning
- Definite Integrals
- Deriving Equations
- Determinant of Inverse Matrix
- Determinants
- Differential Equations
- Differentiation
- Differentiation Rules
- Differentiation from First Principles
- Differentiation of Hyperbolic Functions
- Direct and Inverse proportions
- Disjoint and Overlapping Events
- Disproof by Counterexample
- Distance from a Point to a Line
- Divisibility Tests
- Double Angle and Half Angle Formulas
- Drawing Conclusions from Examples
- Ellipse
- Equation of Line in 3D
- Equation of a Perpendicular Bisector
- Equation of a circle
- Equations
- Equations and Identities
- Equations and Inequalities
- Estimation in Real Life
- Euclidean Algorithm
- Evaluating and Graphing Polynomials
- Even Functions
- Exponential Form of Complex Numbers
- Exponential Rules
- Exponentials and Logarithms
- Expression Math
- Expressions and Formulas
- Faces Edges and Vertices
- Factorials
- Factoring Polynomials
- Factoring Quadratic Equations
- Factorising expressions
- Factors
- Finding Maxima and Minima Using Derivatives
- Finding Rational Zeros
- Finding the Area
- Forms of Quadratic Functions
- Fractional Powers
- Fractional Ratio
- Fractions
- Fractions and Decimals
- Fractions and Factors
- Fractions in Expressions and Equations
- Fractions, Decimals and Percentages
- Function Basics
- Functional Analysis
- Functions
- Fundamental Counting Principle
- Fundamental Theorem of Algebra
- Generating Terms of a Sequence
- Geometric Sequence
- Gradient and Intercept
- Graphical Representation
- Graphing Rational Functions
- Graphing Trigonometric Functions
- Graphs
- Graphs and Differentiation
- Graphs of Common Functions
- Graphs of Exponents and Logarithms
- Graphs of Trigonometric Functions
- Greatest Common Divisor
- Growth and Decay
- Growth of Functions
- Highest Common Factor
- Hyperbolas
- Imaginary Unit and Polar Bijection
- Implicit differentiation
- Inductive Reasoning
- Inequalities Maths
- Infinite geometric series
- Injective functions
- Instantaneous Rate of Change
- Integers
- Integrating Polynomials
- Integrating Trig Functions
- Integrating e^x and 1/x
- Integration
- Integration Using Partial Fractions
- Integration by Parts
- Integration by Substitution
- Integration of Hyperbolic Functions
- Interest
- Inverse Hyperbolic Functions
- Inverse Matrices
- Inverse and Joint Variation
- Inverse functions
- Iterative Methods
- Law of Cosines in Algebra
- Law of Sines in Algebra
- Laws of Logs
- Limits of Accuracy
- Linear Expressions
- Linear Systems
- Linear Transformations of Matrices
- Location of Roots
- Logarithm Base
- Logic
- Lower and Upper Bounds
- Lowest Common Denominator
- Lowest Common Multiple
- Math formula
- Matrices
- Matrix Addition and Subtraction
- Matrix Determinant
- Matrix Multiplication
- Metric and Imperial Units
- Misleading Graphs
- Mixed Expressions
- Modulus Functions
- Modulus and Phase
- Multiples of Pi
- Multiplication and Division of Fractions
- Multiplicative Relationship
- Multiplying and Dividing Rational Expressions
- Natural Logarithm
- Natural Numbers
- Notation
- Number
- Number Line
- Number Systems
- Numerical Methods
- Odd functions
- Open Sentences and Identities
- Operation with Complex Numbers
- Operations with Decimals
- Operations with Matrices
- Operations with Polynomials
- Order of Operations
- Parabola
- Parallel Lines
- Parametric Differentiation
- Parametric Equations
- Parametric Integration
- Partial Fractions
- Pascal's Triangle
- Percentage
- Percentage Increase and Decrease
- Percentage as fraction or decimals
- Perimeter of a Triangle
- Permutations and Combinations
- Perpendicular Lines
- Points Lines and Planes
- Polynomial Graphs
- Polynomials
- Powers Roots And Radicals
- Powers and Exponents
- Powers and Roots
- Prime Factorization
- Prime Numbers
- Problem-solving Models and Strategies
- Product Rule
- Proof
- Proof and Mathematical Induction
- Proof by Contradiction
- Proof by Deduction
- Proof by Exhaustion
- Proof by Induction
- Properties of Exponents
- Proportion
- Proving an Identity
- Pythagorean Identities
- Quadratic Equations
- Quadratic Function Graphs
- Quadratic Graphs
- Quadratic functions
- Quadrilaterals
- Quotient Rule
- Radians
- Radical Functions
- Rates of Change
- Ratio
- Ratio Fractions
- Rational Exponents
- Rational Expressions
- Rational Functions
- Rational Numbers and Fractions
- Ratios as Fractions
- Real Numbers
- Reciprocal Graphs
- Recurrence Relation
- Recursion and Special Sequences
- Remainder and Factor Theorems
- Representation of Complex Numbers
- Rewriting Formulas and Equations
- Roots of Complex Numbers
- Roots of Polynomials
- Rounding
- SAS Theorem
- SSS Theorem
- Scalar Triple Product
- Scale Drawings and Maps
- Scale Factors
- Scientific Notation
- Second Order Recurrence Relation
- Sector of a Circle
- Segment of a Circle
- Sequences
- Sequences and Series
- Series Maths
- Sets Math
- Similar Triangles
- Similar and Congruent Shapes
- Simple Interest
- Simplifying Fractions
- Simplifying Radicals
- Simultaneous Equations
- Sine and Cosine Rules
- Small Angle Approximation
- Solving Linear Equations
- Solving Linear Systems
- Solving Quadratic Equations
- Solving Radical Inequalities
- Solving Rational Equations
- Solving Simultaneous Equations Using Matrices
- Solving Systems of Inequalities
- Solving Trigonometric Equations
- Solving and Graphing Quadratic Equations
- Solving and Graphing Quadratic Inequalities
- Special Products
- Standard Form
- Standard Integrals
- Standard Unit
- Straight Line Graphs
- Substraction and addition of fractions
- Sum and Difference of Angles Formulas
- Sum of Natural Numbers
- Surds
- Surjective functions
- Tables and Graphs
- Tangent of a Circle
- The Quadratic Formula and the Discriminant
- Transformations
- Transformations of Graphs
- Translations of Trigonometric Functions
- Triangle Rules
- Triangle trigonometry
- Trigonometric Functions
- Trigonometric Functions of General Angles
- Trigonometric Identities
- Trigonometric Ratios
- Trigonometry
- Turning Points
- Types of Functions
- Types of Numbers
- Types of Triangles
- Unit Circle
- Units
- Variables in Algebra
- Vectors
- Verifying Trigonometric Identities
- Writing Equations
- Writing Linear Equations
- Statistics
- Bias in Experiments
- Binomial Distribution
- Binomial Hypothesis Test
- Bivariate Data
- Box Plots
- Categorical Data
- Categorical Variables
- Central Limit Theorem
- Chi Square Test for Goodness of Fit
- Chi Square Test for Homogeneity
- Chi Square Test for Independence
- Chi-Square Distribution
- Combining Random Variables
- Comparing Data
- Comparing Two Means Hypothesis Testing
- Conditional Probability
- Conducting a Study
- Conducting a Survey
- Conducting an Experiment
- Confidence Interval for Population Mean
- Confidence Interval for Population Proportion
- Confidence Interval for Slope of Regression Line
- Confidence Interval for the Difference of Two Means
- Confidence Intervals
- Correlation Math
- Cumulative Frequency
- Data Analysis
- Data Interpretation
- Discrete Random Variable
- Distributions
- Dot Plot
- Empirical Rule
- Errors in Hypothesis Testing
- Events (Probability)
- Frequency Polygons
- Generalization and Conclusions
- Geometric Distribution
- Histograms
- Hypothesis Test for Correlation
- Hypothesis Test of Two Population Proportions
- Hypothesis Testing
- Inference for Distributions of Categorical Data
- Inferences in Statistics
- Large Data Set
- Least Squares Linear Regression
- Linear Interpolation
- Linear Regression
- Measures of Central Tendency
- Methods of Data Collection
- Normal Distribution
- Normal Distribution Hypothesis Test
- Normal Distribution Percentile
- Point Estimation
- Probability
- Probability Calculations
- Probability Distribution
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- Quantitative Variables
- Quartiles
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- Randomized Block Design
- Residuals
- Sample Mean
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- Sampling
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- Standard Deviation
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- Stem and Leaf Graph
- Survey Bias
- Transforming Random Variables
- Tree Diagram
- Two Categorical Variables
- Two Quantitative Variables
- Type I Error
- Type II Error
- Types of Data in Statistics
- Venn Diagrams

Have you wondered what the distance between the Earth and the moon was? Can you guess what it might be in meters? Well, based on a study made by NASA, the distance spans approximately 382,500,000 meters. I don't know about you, but that's quite a hefty lot of numbers for me to handle. However, not to worry. This is where scientific notation comes in! It provides us with a method to deal with extremely large numbers or significantly small ones in an easier way.

In this article, we will discuss the concept of a scientific notation. Furthermore, we will become acquainted with a technique that demonstrates how we can convert a number from standard form to its corresponding scientific notation and vice versa. \(x\)

To begin our topic, let us first define the meaning of a scientific notation.

**Scientific notation**, also known as standard form, is a method that expresses (or rewrites) a multi-digit number in a compact way. It takes the form

where $1<=\left|a\right|<10$ and $b$ is an integer.

This is a very effective way to write very large numbers or very small ones as well. It may be helpful to note that for the structure introduced above, that is

$a\times {10}^{b}$

where $1<=\left|a\right|<10$, a is the coefficient and 10 is a constant base. The table below shows several examples where scientific notation takes place.

Number | Scientific notation | a and b values |

$2$ | $2\times {10}^{0}$ | $a=2,b=0$ |

$20$ | $2\times {10}^{1}$ | $a=2,b=1$ |

$200$ | $2\times {10}^{2}$ | $a=2,b=2$ |

What this means is that a larger number can be rewritten in a shorter manner by increasing the power of 10. Or in other words, the value of $b$.

Coincidentally, this works in the reverse too. Numbers that are particularly close to zero can also be expressed in this way by changing the sign of the exponent. Here is a table that demonstrates a few examples.

Number | Scientific notation | $a$ and $b$ values |

$0.2$ | $2\times {10}^{-1}$ | $a=2,b=-1$ |

$0.02$ | $2\times {10}^{-2}$ | $a=2,b=-2$ |

$0.002$ | $2\times {10}^{-3}$ | $a=2,b=-3$ |

A number can be written in scientific notation by expressing the value as **a number between 1 and 10 multiplied by a power of 10**. For example, the number 700 can be written as $7\times {10}^{2}$. The number $7\times {10}^{2}$ then is the scientific notation of 700.

The format for scientific notation is $a\times {10}^{b}$ where

**a**is a number or decimal number such that the absolute value of**a**is**greater than**or**equal**to**one**and**less than ten**and;**b**is the**power of 10**required so that the scientific notation is mathematically equivalent to the original number.

To write a number in scientific notation, one must adhere to the following rules:

The base is always 10.

The value of the coefficient is always greater or equal to 1 and less than 10.

The coefficients can also be either positive or negative values.

The rest of the significant digits of the number is carried by the mantissa.

The

is the part of a logarithm after the decimal point.*mantissa*

In this section, we will take a look at a worked example involving scientific notation.

- $650,000,000=6.5\times {10}^{8}$
- $75=7.5\times {10}^{1}=7.5\times 10$
- $5.05\times {10}^{7}$
- $0.00001=1\times {10}^{-5}$
- $1230000000=1.23\times {10}^{9}$

Notice that all the conditions for writing scientific notations are met in the example above:

- the base of each example is 10;
- the coefficients are greater or equal to 1 and less than 10;
- and the exponent of the base is accounted for the mantissa.

Although the method behind scientific notation can be straightforward, there are still some common errors you should consider so that you don't fall trap of making careless mistakes in your work. Here is an example where scientific notations are not valid.

$76400=76.4\times {10}^{3}$

This is incorrect because the$76400=7.64\times {10}^{4}$

Here is another worked example.

$160=2.5\times {8}^{2}$

Although this equation may be true, it is not a valid scientific notation. Notice the base used in the example above. Recall that$160=1.6\times {10}^{2}$

Let's look at one more example before we move on to our next section.

$0.034=34\times {10}^{-3}$

As before,$0.034=3.4\times {10}^{-2}$

In this section, we will be learning how to interchange a given number between its standard form and scientific notation.

To understand how we can convert numbers in standard form to their appropriate scientific notation, we shall demonstrate two cases for you to consider.

This means that the power of 10 here will be a positive value. Here is an example.

The population of the world is currently at 7,000,000,000. To express this in scientific notation, we can write this as

$7\times {10}^{9}$.

In this instance, the power of 10 will become a negative value. Here is an example.

If we have to write the diameter of a grain of sand, which is 24 ten-thousandths inches or .0024 inch, we would get

$0.0024=2.4\times {10}^{-3}$.

There is no particular rule to follow when converting a number in scientific notation to standard form. However, there are a few pointers we should note when doing so.

Move the decimal point to the

**right**if the exponent of the base is**positive**.Move the decimal point to the

**left**if the exponent of the base is**negative**.Move the decimal point

**as many times**as indicated by the**exponent**.In standard form, do not write multiply by 10 anymore.

Let us take a look at a few examples to see how this is done.

The scientific notation of the distance from the Earth to the moon is $3.825\times {10}^{8}$ meters. How would you represent this number in standard form?

**Solution**

Since the exponent of the base is positive, move the decimal point to the right. By the third movement, we should be at 3825. This means any movement after this adds a 0 to the figure. This will add 5 more zeros to the number. Thus resulting in 382,500,000 meters.

Here is another example.

The length of the shortest wavelength of visible light is considered to be $4.0\times {10}^{-7}$meters. Write this in standard form.

**Solution**

The decimal point will be moved to the left since the exponent of the base is negative. One move of the decimal point will leave us at 0.4. However, we need to do all 7 movements. Each one after that will add a zero before 4. Hence, we will have 0.0000004 meters.

In this segment, we will discuss how we can perform basic arithmetic operations with numbers in scientific notation. It can get rather complex and confusing when dealing with extremely large or significantly small numbers. The purpose of scientific notation is to make numbers easier to read, write, and calculate. Numbers in scientific notation can be added, subtracted, multiplied, and divided while they are still in scientific notation.

Below are steps to add and subtract numbers in scientific notation.

Make both numbers you are attempting to add or subtract have

**the same exponent**by rewriting the number with the smaller exponent and moving the decimal point to its decimal number the required number of times.Add or subtract these decimal numbers.

Write your number in scientific notation if necessary.

Here is an example that demonstrates this.

$(6.7\times {10}^{4})+(5.87\times {10}^{5})$.

Rewriting the number will have us with

$(0.67\times {10}^{5})+(5.87\times {10}^{5})$

We will now have;$(0.67+5.87)\times {10}^{5}$.

With our example, we will have;

$6.54\times {10}^{5}$

In this section, we will observe a real-world problem that makes use of adding and subtracting numbers in scientific notation.

Amy travels $2.33\times {10}^{8}$meters from her home to her workplace. After work, she was told to attend a meeting in town. From her workplace, she travels $8.2\times {10}^{9}$meters to the venue of this meeting. Find the total distance she covered today.

**Solution**

This problem is asking you to add the two given numbers in scientific notation, namely $2.33\times {10}^{8}$ and $8.2\times {10}^{9}$ since we are looking for her total journey. In doing so, we can write this as

$(2.33\times {10}^{8})+(8.2\times {10}^{9})$Rewrite both to have the same exponent.

$(0.233\times {10}^{9})+(8.2\times {10}^{9})$

Add the decimals

$(0.233+8.2)\times {10}^{9}$

Thus, we have $8.433\times {10}^{9}$ meters in total.

Below are steps to multiply or divide numbers in scientific notation.

Multiply or divide the decimal numbers.

Now either multiply by adding exponents or divide by subtracting the exponents of the 10.

Write your answer in scientific notation if necessary.

Here is a worked example.

Solving $(8.4\times {10}^{-3})(6.1\times {10}^{6})$, we will have

$8.4\times 6.1=51.24$

Then,

${10}^{-3}\times {10}^{6}={10}^{-3+6}={10}^{3}$We will now have

$51.24\times {10}^{3}$

In this section, we will observe a real-world problem that makes use of multiplying and dividing numbers in scientific notation.

Given the perimeter of a rectangle to be $6\times {10}^{7}$ , and its length to be $8\times {10}^{5}$, find its width.

**Solution**

If $Perimeter=Length\times Width$ then rearranging this will become

$Width=\frac{Perimeter}{Length}$

Divide the decimal numbers.

$6\xf78=0.75$

Subtract exponents of 10.

${10}^{7-5}={10}^{2}$

$0.75\times {10}^{2}$

Following the rules of scientific notation, the coefficient needs to be between 1 and 10. Hence, this can be worked on more by moving the decimal point to the right in 1 decimal place. Moving the decimal point to the right by 1 reduces the exponent of its base by 1 also.

$Width=7.5\times {10}^{1}cm$

Multiplying scientific notation is finding the product of their coefficients and adding their exponents. In this regard, dividing them is also equivalent to finding their quotient and subtracting their exponents.

- Scientific notation is a way to rewrite multi-digit numbers in a compact way in the form $a\times {10}^{b}$ where $1<=\left|a\right|<10$ and $b$ is an integer.
- The value of the coefficient is always greater or equal to 1 and less than 10.
- The base in scientific notation is always 10.
- When adding or subtracting numbers in scientific notation, be sure all exponents involved have the same value.
- When multiplying in scientific notation, multiply the coefficients and add the exponents of the base.
- When dividing in scientific notation, divide the coefficients and subtract the exponents of the base.

Scientific notation is a symbolic system of representation of exceptionally complex numbers.

- The base is always 10.
- The exponent can be a negative or positive value but not 0.
- The value of the coefficient is always greater or equal to 1 and less than 10.
- The coefficients can also be either positive or negative values.
- The rest of the significant digits of the number is carried by the mantissa.

The scientific notation for 700 is 7 ✕ 10^2

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