126 LearnersLast updated on July 22nd, 2025
Powers of 10
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.
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”.
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) |
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.
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 = 107
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
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”.
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
Misinterpreting Negative Exponents
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
Confusion Between Base and Exponent
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
Forgetting to Find the Common Factor in Addition and Subtraction
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
Forgetting the Rule of Multiplying Numbers with Powers of 10
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
Incorrect Division of Powers of 10
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.
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.
Solved Examples of Powers of 10
Problem 1
Add 10^4 + 10^9
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
Problem 2
Subtract 10^5 - 10^2
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
Problem 3
Multiply 10^5 × 10^9
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
Problem 4
Divide 10^8 / 10^5
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.
Problem 5
Divide 10^-9 / 10^-16
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.
FAQs on Powers of 10
1.What do you mean by powers of 10?
2.How much is 10^8?
3. How to write 1000 as a power of 10?
4. What is the rule for multiplying numbers with powers of 10?
5.Define the rule for dividing numbers with powers of 10.
6.How can children in Vietnam use numbers in everyday life to understand Powers of 10?
7.What are some fun ways kids in Vietnam can practice Powers of 10 with numbers?
8.What role do numbers and Powers of 10 play in helping children in Vietnam develop problem-solving skills?
9.How can families in Vietnam 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.
Fun Fact
: She loves to read number jokes and games.
