SweetHaven Pre-Algebra Prime, Version 4.1

Chapter 5—Powers, Exponents, and Roots
5-4 Powers of Ten

When you complete the work for this section, you should be able to:

Rewrite any power of ten as an ordinary decimal value.
Rewrite decimal values with power-of-10 notation

Your success with power-of-10 notation depends entirely upon your understanding of exponents that have a base of 10.

First, recall that powers are simply convenient ways to express how a number is multiplied by itself, over and over again:

10² = 10 • 10 = 100 (ten multiplied by itself 2 times)
10³ = 10 • 10 • 10 = 1000 (ten multiplied by itself 3 times)
10¹² = ten multiplied by itself 12 times, or 1000000000000 (ten multiplied by itself 12 times)

Next, there are two special cases — memorize them:

10¹ = 10
10⁰ = 1

Finally, there are the negative exponents:

10^-1 = 1/10 = 0.1
10^-2 = 1/10² = 1/100 = 0.01
10^-3 = 1/10³ = 1/1000 = 0.001
10^-12 = 1/1000000000000 = 0.000000000001

Important

A negative exponent does not mean the decimal value is negative. It means the decimal value is a fractional decimal — a value less than 1.

Use this button to open a table of powers of ten. You will find a wide range of powers-of-10 expressed in terms of their fraction and decimal values. Make sure you understand exactly what this table means. If you know what it means, you don't have to memorize it ... you will know how to figure out the values for yourself.

Writing Powers of 10 as Decimal Values

Mental Shortcut

Mental shortcuts are not real mathematical principles. They are merely tricks we used to simplify and speed up the process of solving problems. Such shortcuts should never be confused with real mathematical principles and processes.

When the exponent of a power-of-10 expression is a positive integer:

10¹ = 10, or 1 with the decimal point moved one place to the right
10² = 100, or 1 with the decimal point moved two places to the right
10¹⁸ represents 1 followed by 18 zeros.

Example

Problem: Rewrite 10⁴ in decimal form.

Procedure:

Solution: 10⁴ = 10000

When the exponent of a power-of-10 expression is a negative integer:

10^-1 = 0.1, or 1 with the decimal point moved one place to the left
10^-2 = 0.01, or 1 with the decimal point moved two places to the left
10^-18 represents 1 preceded by 17 zeros and a decimal point.

Example

Problem: Rewrite 10^-4 in decimal form

Procedure:

Solution: 10^-4 = 0.0001

Examples and Exercises

Rewrite these powers of ten as decimal values.

Rewriting Decimals as Powers of Ten

Earlier in this lesson, you saw that 10³ = 1000. This can be stated the other way around: 1000 = 10³. Also:

1 = 1 x 10⁰
10 = 1 x 10¹
0.1 = 1 x 10^-1
0.0001 = 1 x 10^-4

Mental Shortcut

Mental shortcuts are not real mathematical principles. They are merely tricks we used to simplify and speed up the process of solving problems. Such shortcuts should never be confused with real mathematical principles and processes.

When a decimal value is greater than 1:

10 = 1 when the decimal point is moved one place to the left
100 = 1 when the decimal point moved two places to the left
1 000 000 000 = 1 when the decimal point is moved nine places to the left

Example

Problem: Rewrite 10000 as a power of 10

Procedure:

Solution: 10000 = 10⁴

When a decimal value is less than one:

0.01 = 1 when the decimal point is moved two places to the right
0.000001 = 1 when the decimal point is moved six places to the right
0.000 000 001 =1 when the decimal point is moved nine places to the right

Example

Problem: Rewrite 0.0001 as a power of 10

Procedure:

Solution: 0.0001 = 10^-4

Examples and Exercises

Rewrite these decimals as powers of ten.