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# Algebraic Exponents to the Power of 10

by Ron Kurtus (revised 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 =10^{6}

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 **10 ^{13}**.

## 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 = 1/10^{2}

1/10^{2}=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*10 ^{6}** 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*10 ^{7}** miles. That is the more accepted format.

Likewise,

12,300,000 =

123*10^{5}=

1.23*100*10^{5}=1.23*10.^{7}

### 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*10 ^{7}** might be displayed as

**9.2E7**. Likewise,

**2.5*10**would display as

^{−3}**2.5E−3**on the calculator.

## Summary

You can denote large numbers that are multiples of **10** as powers, such as **1,000,000 = 10 ^{6}**. You can also denote small numbers as multiples of

**1/10**as powers of

**10**, such as

**0.0001 = 10**. This leads to a convenient way to denote very large or very small numbers that include other digits than

^{−4}**10**, such as

**1.3*10**to denote

^{4}**13,000**. Calculators often use the variation of denoting the power of

**10**, such as

**2E3**to denote

**2*10**.

^{3}Be clever in the way you do things

## Resources and references

### Websites

**Exponents: Basic Rules** - PurpleMath.com

**Exponent Rules** - RapidTables.com

**Laws of Exponents** - MathisFun.com

**Exponents Calculator** - CalculatorSoup.com

### Books

(Notice: The *School for Champions* may earn commissions from book purchases)

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exponents_power_10.htm**

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## Algebraic Exponents to the Power of 10