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Last updated on July 6th, 2025

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Powers of 10

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Numbers with a base of 10 and an exponent that is an integer are known as the powers of 10. When 10 is multiplied by itself a certain number of times, we can represent the result using an exponent. In this article, we will explore numbers with powers of 10 in detail.

Powers of 10 for Filipino Students
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What does Powers of 10 Mean?

When 10 is multiplied by itself several times, the result can be written using exponents - these are called the powers of 10. For example, if we multiply 10 eight times, the product is 100,000,000 (10 × 10 × 10 × 10 × 10 × 10 × 10 × 10). In this case, we can use the exponents and simplify it as 108. This means 10 is multiplied 8 times, with the base being 10 and 8 being the exponent. This is referred to as “10 to the eighth power”.

 

Professor Greenline from BrightChamps

What are Positive and Negative Powers?

Positive powers of 10 make the number larger by adding zeros, while negative powers make the number smaller by turning it into a decimal. 


Positive powers of 10:

If the exponent is a positive integer, multiplying 10 by itself results in a large number. We raise 10 to different powers according to their exponents. For example, 103 is read as “10 to the power of three” and called a thousand (1,000). The letter ‘k’ represents it, and the SI prefix is known as “kilo”.

Likewise, some powers of 10 have specific names and symbols, which are listed below:     

 

 

Positive Powers of 10 Name Prefix (symbol)
101 = 10 Ten

Deca (D)

102 = 100 Hundred Hecto (H)
103 = 1000 Thousand Kilo (K)
106 = 1000000 Million Mega (M)
109 = 1000000000 Billion Giga (G)
1012 = 1000000000000 Trillion Tera (T)
1015 = 1000000000000000 Quadrillion Peta (P)
1018 =  1000000000000000000 Quintillion Exa (E)
1021 = 1000000000000000000000 Sextillion

Zetta (Z)

1024 =  1000000000000000000000000 Septillion Yotta (Y)




Negative powers:

If the exponent is a negative number, multiplying 10 by itself results in a very small number. We use a different method to solve negative powers. If the exponent is a negative number, we find the reciprocal of the base and then solve it like a positive. This is called the multiplicative inverse of the base. For example, we have (3/5)-2  and it can be rewritten as (5/3)2


Likewise, negative powers of 10, such as 10-4 become:
 1/104 or 1/(10 × 10 × 10 × 10) = 1/10000 = 0.0001.  
Hence, 10-4 gives a small number, which is less than 1. 


Here are some names and symbols for the negative powers of 10: 



 

Negative Powers of 10 Name Prefix (Symbol)
10-1 = 0.1 Tenth Deci (d)
10-2 = 0.01 Hundredth Centu (c)
10-3 = 0.001 Thousandth Milli (m)
10-6 = 0.000001 Millionth Micro  (μ)
10-9 = 0.000000001 Billionth Nano (n)
10-12 = 0.000000000001 Trillionth Pico (p)
1015 = 0.000000000000001 Quadrillionth Femto (f)
10-18 = 0.000000000000000001 Quintillionth Atto (a)
10-21 = 0.000000000000000000001 Sextillionth  Zepto (z)
10-24 = 0.000000000000000000000001 Septillionth Yocto (y)

 

Professor Greenline from BrightChamps

How to Use Powers of 10?

Now, let us look at how to represent a number as a power of 10. For instance, take the number 1,000. 
 

Step 1: Break the number down using multiples of 10. 
1,000 = 10 × 10 × 10 

Step 2: Find how many times 10 has been multiplied by itself. 
Here, 10 is multiplied by three times. So, there are three 10s.     

Step 3: Represent the number as a power of 10. 
1,000 = 103  

A number with a power of 10 can be solved as:

For example, solve 35 × 103.

103 = 10 × 10 × 10 = 1000
35 × 103 = 35 × 1000 = 35,000

The digits are moved to the right when multiplying a number by powers of 10. In this case, we shift three places to the left when multiplying it by 103
 

For instance, solve 420 ÷ 102

102 = 10 × 10 = 100
420 ÷ 100 = 4.2 

When dividing a number by powers of 10, the decimal point moves to the left. Here, we shift two places to the left when dividing it by 102.  
 

Professor Greenline from BrightChamps

Calculating Powers of 10

We should first find out the values of powers of 10 to perform addition, subtraction, multiplication, and division of numbers with powers of 10. 

 

Addition and subtraction of powers of 10:

Before performing addition and subtraction, we need to identify a common factor between two numbers. The number with the smallest exponent will be the common factor. After that, simplify the remaining number then perform the respective operations, and multiply back by the common factor. 

For example, find the sum of 106 and  109.

Step 1: Find the smallest exponent. 
6 is the smallest exponent because 106 is smaller than 109

Step 2: Since 106 is the common factor, we can factor out 106 from the expression:
106 + 109 = 106 × (1 + 103)

When you factor out 106, you are left with 1 from the first term:
     106 ÷ 106 = 1 
From the second term, 109 ÷ 109, so you are left with 103.


Step 3: Simplify the expression. 
    1 + 103 = 1 + 10 × 10 × 10 = 1 + 1000 = 1001


Step 4: Multiply the result by 106
 106 × 1001 
106 = 10 × 10 × 10 × 10 × 10 × 10 = 1,000,000 
1,000,000 × 1001 = 1,001,000,000 

Thus, 106 + 109 = 1,001,000,000

 

Next, the subtraction of numbers with powers of 10.

For example, 105 - 102. 


Step 1: Find the smallest powers of 10. 
Here, 102 is smaller than 105.  
 

Step 2: 102 is the common factor, so factor it out. 
105 - 102 = 102 × (103 - 1) 

After factoring out 102:
From 102, we get 102 ÷ 102 = 1. 
From 105,  we get 105 ÷ 102 =  103
Hence, the expression becomes 103 - 1.  


Step 3: Simplify the expression. 
103 - 1 = 10 × 10 × 10 = 1000 
1000 - 1 = 999 
  
Step 4: Multiply back the common factor 102.
102 × 999 
10 × 10 = 100
100 × 999 = 99,900

Thus, 105 - 102 = 99,900. 
 

 

Multiplying powers of 10:

When we multiply numbers with powers of 10 if the bases are the same, we add the exponents. The rule is:
am × an = am + n

For example, 103 × 104

Here, the bases are 10. Hence, we simply add the exponents. 
103 × 104 = 103 + 4 = 10

Now, we find the value of the power of 10. 
107 = 10 × 10 × 10 × 10 × 10 × 10 × 10 = 10,000,000 

Thus, 107 = 10,000,000
 

Let us take another example.

Multiply 10-2 × 105

The bases are 10, so we need to add the exponents. 
10-2 × 105 = 10-2 + 5 = 103
Hence, 10-2 × 105 = 103

Next, we can find the value of 103.
103 = 10 × 10 × 10 = 1,000 

Therefore, 103 = 1,000 


 

Dividing powers of 10:

While dividing numbers, if the bases are the same, we need to subtract the exponents. The rule states that, 
   am / an = am - n

 For example, 106 / 104.

The bases are the same, so we need to subtract the exponents. 

106 / 104 = 106 - 4 = 102 
106 / 104 = 102 

Next, find the value of 102
102 = 10 × 10 = 100
 
Thus, 102  = 100.


Let us take another example,
divide 10-8 / 10-14.

10- 8 / 10-14 = 10-8 - (-14) 
10- 8 + 14  = 106

106 = 1,000,000 
 

 

Professor Greenline from BrightChamps

Tips and Tricks

When working with numbers involving powers of 10, we should keep some tips and tricks in mind to solve mathematical problems efficiently and accurately. 

  • Numbers with powers of 10 have two parts: a base and an exponent. The exponent represents the number of times 10 is multiplied by itself.

    For example, 104 means:

     10 × 10 × 10 × 10 = 10,000.
    So, 104 equals 10,000.  

     
  • Remember, 410, which means “4 to the power of 10”, is different from 104, which means “10 to the power of 4”. 
Max Pointing Out Common Math Mistakes

Common Mistakes and How to Avoid Them on Powers of 10

Powers of any number involve silly mistakes. Students often make mistakes in calculating the accurate values. Here are some common mistakes and their helpful solutions to prevent these errors. 

Mistake 1

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Misinterpreting Negative Exponents

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Students mistakenly assume that a negative exponent makes the answer negative,  which will lead them to wrong answers. If the exponent is negative, take the reciprocal of the base and then solve it just like a positive exponent.  


For instance, 10-3 can be written as:

10-3 = 1/103
Now calculate 103: 10 × 10 × 10 = 1000 

Hence, = 1 / 10-3 = 1 / 1000 = 0.001
 

Mistake 2

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Confusion Between Base and Exponent 

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Sometimes, students mix up the base and exponent, which can lead to incorrect answers. Always remember that the number that gets multiplied is known as the base, and the exponent explains how many times the base is multiplied by itself. 


For example, students might incorrectly interpret 105 as 10 × 5 = 50. 

The correct interpretation is that 10 is multiplied by itself five times:
10 × 10 × 10 × 10 × 10 = 100,000. 
 

Mistake 3

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 Forgetting to Find the Common Factor in Addition and Subtraction 

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Always remember to identify the common factor before addition and subtraction. It is the smallest exponent in the given numbers. Otherwise, the final sum or difference will be incorrect. 

For instance, add 103  + 108.
Here, the smallest exponent is 3, so the common factor is 103.
Next, factor out 103
103 + 108 = 103 (1 + 105)   
 

Mistake 4

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Forgetting the Rule of Multiplying Numbers with Powers of 10 

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 Keep in mind that when multiplying powers of 10 with the same bases, add the exponents. If students fail to remember the rule, they might mistakenly multiply the exponents or the bases incorrectly.

The exponent  rule is:
 am × an = am + n


For instance, 106 × 103 = 106 + 3 = 109.
This means:
 1,000,000 × 1,000 = 1,000,000,000

Mistake 5

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 Incorrect Division of Powers of 10 

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Students should remember the rule for division:  am / an = am - n when dividing powers of 10. This rule states that subtract the exponents when the bases are the same. 

For example, 105 / 103 = 105 - 3 
= 102 = 100.
 

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Professor Greenline from BrightChamps

Real-Life Applications of Powers of 10

Learning the significance of the powers of 10 will help us easily apply them to various real-life situations. Here are some real-world applications of powers of 10. 
 

  • Astronomers and aerospace researchers use the powers of 10 to measure the distance between different celestial bodies.

    For example, the distance between the Earth and the Moon is difficult to write - about 760,000 kilometers.

    Hence, professionals use the powers of 10 and express it in scientific notation as 7.6 × 105 kilometers, which makes it easier to write and understand.  

     
  • In scientific research laboratories, scientists can use the powers of 10 to express very small values.

    For instance, the diameter of a typical plant or animal cell is 10-6 meters. 

     
  • Governments can use the powers of 10 to represent the population count of a country or the world.

    For example, if an organization wants to estimate the population of a city, it can be represented as 1.5 × 109 instead of 1,500,000,000. 
     
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Solved Examples of Powers of 10

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Max, the Girl Character from BrightChamps

Problem 1

Add 10^4 + 10^9

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 1,000,010,000

Explanation

To find the sum of 104 and 109, we need to find the smallest exponent. 
Here, 4 is the smallest exponent because 104 is smaller than 109

Therefore, 104 is the common factor, and we must factor out 104
104 + 109 = 104 × (1 + 105)

After factoring out 104, 1 is remaining from 104, and 105 is remaining from 109

Next, simplify the expression inside the parentheses. 
1 + 105 = 1 + 10 × 10 × 10 × 10 × 10  = 1 + 100,000 = 100,001 

Finally, we can multiply the result by 104
 104 × 100,001 
 104 = 10 × 10 × 10 × 10 = 10,000.
 10,000 × 100,001 = 1,000,010,000 

Therefore,  104 + 109 = 1,000,010,000  
 

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Max, the Girl Character from BrightChamps

Problem 2

Subtract 10^5 - 10^2

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99,900

Explanation

First, we need to find the smallest power of 10. 

Here, 102 is smaller than 105.

Hence, 102 is the common factor, so factor it out.
 105 - 102 = 102 × (103 - 1)

After factoring out 102, 1 is remaining from 102 and 103 is remaining from 105
103 - 1 = (10 × 10 × 10) - 1 = 1000 - 1 = 999

Then, multiply back the common factor 102

Now, expand 102: 10 × 10 = 100 

Next, multiply 100 and 999:
 100 × 999 = 99,000

Therefore, 105 - 102 = 99,900   
 

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Max, the Girl Character from BrightChamps

Problem 3

Multiply 10^5 × 10^9

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100,000,000,000,000
 

Explanation

The rule for multiplying numbers with powers of 10 is:
am × an = am + n

Here, the bases are the same (both are 10). 
105 × 109 = 105 + 9 = 1014

Next, we can find the value of 1014

1014 = 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 

1014 = 100,000,000,000,000

Thus, 105 × 109 = 100,000,000,000,000

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Max, the Girl Character from BrightChamps

Problem 4

Divide 10^8 / 10^5

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1,000

Explanation

Here, the bases are the same (both are 10). 

The rule for dividing powers of 10 is:
am / an = am - n

108 / 105 = 108 - 5 = 103

Next, find the value of 103.
103 = 10 × 10 × 10 = 1,000.

Therefore, 108 / 105 = 1,000.

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Max, the Girl Character from BrightChamps

Problem 5

Divide 10^-9 / 10^-16

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 10,000,000

Explanation

The rule is:
am / an = am - n

We can apply the rule by subtracting the exponents.
 10-9 / 10-16 = 10-9 - (-16)
 10-9 + 16 = 107

Now, we can find the value of 107.

107 = 10 × 10 × 10 × 10 × 10 × 10 × 10 = 10,000,000

Hence, 10-9 / 10-16 = 10,000,000. 
 

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FAQs on Powers of 10

1.What do you mean by powers of 10?

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2.How much is 10^8?

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3. How to write 1000 as a power of 10?

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4. What is the rule for multiplying numbers with powers of 10?

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5.Define the rule for dividing numbers with powers of 10.

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6.How can children in Philippines use numbers in everyday life to understand Powers of 10?

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7.What are some fun ways kids in Philippines can practice Powers of 10 with numbers?

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8.What role do numbers and Powers of 10 play in helping children in Philippines develop problem-solving skills?

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9.How can families in Philippines create number-rich environments to improve Powers of 10 skills?

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Hiralee Lalitkumar Makwana

About the Author

Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.

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Fun Fact

: She loves to read number jokes and games.

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