logo Your Math Help is on the Way!

More Math Help

Algebraic Symmetries
Radical Expressions and Equation
The Exponential Function
Math 1010-3 Exam #3 Review Guide
Rational Numbers Worksheet
Are You Ready for Math 65?
Solving Simultaneous Equations Using the TI-89
Number Theory: Fermat's Last Theorem
Course Syllabus for Intermediate Algebra
Solving Inequalities with Logarithms and Exponents
Introduction to Algebra Concepts and Skills
Other Miscellaneous Problems
Syllabus for Calculus
Elementary Linear Algebra
Adding and Subtracting Fractions without a Common Denominator
Pre-Algebra and Algebra Instruction and Assessments
Mathstar Research Lesson Plan
Least Common Multiple
Division of Polynomials
Counting Factors,Greatest Common Factor,and Least Common Multiple
Real Numbers, Exponents and Radicals
Math 115 Final Exam Review
Root Finding and Nonlinear Sets of Equations
Math 201-1 Final Review Sheet
Powers of Ten and Calculations
Solving Radical Equations
Factoring Polynomials
Section 8
Declining Price, Profits and Graphing
Arithmetic and Algebraic Structures
Locally Adjusted Robust Regression
Topics in Mathematics
Syllabus for Mathematics
The Quest To Learn The Universal Arithmetic
Solving Linear Equations in One Variable
Examples of direct proof and disproof
Algebra I
Quadratic Functions and Concavity
More on Equivalence Relations
Solve Quadratic Equations by the Quadratic Formula
Solving Equations and Inequaliti
MATH 120 Exam 3 Information
Rational Number Ideas and Symbols
Math Review Sheet for Exam 3
Linear Algebra Notes
Factoring Trinomials
Math 097 Test 2
Intermediate Algebra Syllabus
How to Graphically Interpret the Complex Roots of a Quadratic Equation
The General, Linear Equation
Written Dialog for Problem Solving
Radian,Arc Length,and Area of a Sector
Internet Intermediate Algebra
End Behavior for linear and Quadratic Functions
Division of Mathematics
161 Practice Exam 2
General linear equations
Algebraic Symmetries
Math 20A Final Review Outline
Description of Mathematics
Math 150 Lecture Notes for Chapter 2 Equations and Inequalities
Course Syllabus for Prealgebra
Basic Operations with Decimals: Division
Mathematics Content Expectations
Academic Systems Algebra Scope and Sequence
Syllabus for Introduction to Algebra
Syllabus for Elementary Algebra
Environmental Algebra
More Math Practice Problems
Intermediate Algebra
Syllabus for Linear Algebra and Differential Equations
Intermediate Algebra
Rational Expressions and Their Simplification
Course Syllabus for Intermediate Algebra
GRE Review - Algebra
Foundations of Analysis
Finding Real Zeros of Polynomial Functions
Model Academic Standards for Mathematics
Study Guide for Math 101 Chapter 3
Real Numbers
Math 9, Fall 2009, Calendar
Final Review Solutions
Exponential and Logarithmic Functions

Try the Free Math Solver or Scroll down to Tutorials!












Please use this form if you would like
to have this math solver on your website,
free of charge.

Powers of Ten and Calculations

Power of ten is the mathematical shorthand for writing very large or very small numbers, using positive
or negative exponents of base 10. The exponent is written as a superscript numeral to the right of the 10.

The Basics:

(1) Ten with a positive exponent means to multiply 10 by itself the number of times indicated by the
power. For example, 103 = 10 x 10 x 10 = 1000.

(2) Ten to a negative power means to multiply 0.1 (1/10) by itself the number of times indicated by
the power. For example, 10-3 = 0.1 x 0.1 x 0.1 = 0.001 (equivalent to 1/1000, or the reciprocal of
103, which is 10-3).

(3) Ten to the zero power (100) equals 1, by definition. Zero (0) can only be represented by itself.

(4) Any number can be expressed as a power of ten by using a scalar value and an exponent. For
example, 212 = 2.12 x 100 = 2.12 x 102 and 0.0098 = 9.8 x 10-3.

(5) When multiplying two numbers expressed as powers of ten, add the exponents. For example,

(6) When dividing two numbers expressed as powers of ten, subtract the exponents. For example,

(7) When multiplying or dividing numbers in scientific notation, multiply (or divide) the scalar
values separately from the exponents, then combine the two.
For example,

(8) When adding or subtracting numbers expressed in powers of ten, transform the values to a
common exponent, then add or subtract the scalar (non-exponent) values. For example, 3.8 x 103

To make thinking in large and small numbers a little easier, scientists and engineers commonly
express powers of ten in multiples of 3 (or –3), which is equivalent to counting, 1 thousand, 1 million, 1
billion, 1 trillion, and so on. In computer lingo, this idea is expressed by the prefixes kilo-, mega-, giga-,
and tera-, as in “megabytes.”

Equivalent value Equivalent value

Other things to keep in mind about math with exponents:

Use a leading zero in the ones place for any decimal values
for example, 0.025 instead of .025

When using scientific notation, express the scalar value with a numeral in the ones place
for example, 1.25 x 105 instead of 0.0125 x 107.

The exception to the above guideline is when you need to use a common exponent for
operations such as addition and subtraction;
for example,
instead of

Be careful about moving the decimal point and the exponent in the correct direction
(especially for negative exponents). In your mind (or on paper), convert the number
expressed in scientific notation into a decimal value, and ask yourself whether the
exponent gets larger or smaller. (Admittedly, this can be tricky.)

For example,

0.254 x 102 (0.254 x 100 = 25.4) becomes 2.54 x 101, NOT 2.54 x 103 (which is 2540)
25.4 x 102 (25.4 x 100 = 2540) becomes 2.54 x 103, NOT 2.54 x 101 (which is 25.4)

0.15 is equivalent to 1.5 x 10-1 or 0.015 x 101 (moving the decimal point either way)

A rule of thumb for decimal places and exponents:
For positive exponents, 1 x 10n n is equal to the number of zeroes to the right of 1.

For negative exponents, 1 x 10-n there are (n-1) zeroes to the left of 1
(i.e., the equivalent decimal position would be to the right of the 1)

Remember the relationship between positive and negative exponents and reciprocals of


AND, when adding or subtracting with scientific notation, always remember to convert to
a common exponent:

which is

Express each number as a power of ten:

(can zero be represented as a power of ten?)
Express each as a non-“power of ten” number:

Complete each calculation, show intermediate steps when appropriate, and express the
answer as a power of ten: