# Algebraic Exponents to the Power of 10

by Ron Kurtus (updated 18 January 2022)

You can denote multiples of 10 as a power of 10. You can also denote multiples of 1/10 as powers of 10. This leads to a convenient way to denote very large or very small numbers that include other digits than 10. Calculators often use a variation of denoting the power of 10.

Questions you may have include:

• What are powers of 10?
• How can multiples of 1/10 be written as powers of 10?
• How are other large or small numbers written in powers of 10?

This lesson will answer those questions.

## Large numbers

Large multiples of 10 numbers can be written in terms of powers of 10. For example

1,000,000 =

10*10*10*10*10*10 = 106

This is very convenient, because all you have to do is to add the number of zeros and use that as the exponent.

The number 10,000,000,000,000 has thirteen zeros and thus equals 1013.

## Numbers smaller than 1

Numbers smaller than 1 can be written in powers of 10 by using a negative exponent, which means the reciprocal of the number. For example:

0.01 = 1/100

1/100 = 1/102

1/102 = 10−2

The easy way to figure it out is to count the number of placed to the right of the decimal point. Another example is 0.0001 = 10−4, since there are four places to the right of the decimal point.

## Other numbers

Other large and small numbers can be written as powers of 10.

### Large numbers

Since most large numbers have other digits involved, you can break the number into parts. For example, the distance from the Earth to the Sun is approximately 92,000,000 miles. You could rewrite that number as 92*1000000 miles, which can be designated as 92*106 miles.

### Convention

The usual convention in writing a number times a power of 10 is to reduce the digits in the first number to a value between 1 and 10. This is also called the scientific notation of an exponential number.

Since 92 = 9.2*10, we would write 92,000,000 miles as 9.2*107 miles. That is the more accepted format.

Likewise,

12,300,000 =

123*105 =

1.23*100*105 = 1.23*107.

### Small numbers

You can also denote number with other digits in powers of 10, such as

0.0025 =

25*0.0001 = 25*10−4

Following the convention for writing numbers to the power of 10, you should change 25*10−4 to 2.5*10*10−4 = 2.5*10−3.

### Calculator notation

Most hand-held calculators denote raising to a power of 10 with an E (meaning exponential). In other words, on a calculator 9.2*107 might be displayed as 9.2E7. Likewise, 2.5*10−3 would display as 2.5E−3 on the calculator.

## Summary

You can denote large numbers that are multiples of 10 as powers, such as 1,000,000 = 106. You can also denote small numbers as multiples of 1/10 as powers of 10, such as 0.0001 = 10−4. This leads to a convenient way to denote very large or very small numbers that include other digits than 10, such as 1.3*104 to denote 13,000. Calculators often use the variation of denoting the power of 10, such as 2E3 to denote 2*103.

Be clever in the way you do things

## Resources and references

Ron Kurtus' Credentials

### Websites

Exponents: Basic Rules - PurpleMath.com

Exponent Rules - RapidTables.com

Laws of Exponents - MathisFun.com

Exponents Calculator - CalculatorSoup.com

Algebra Resources

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exponents_power_10.htm

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