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287 LearnersLast updated on December 13, 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 this concept in detail.
What does Powers of 10 Mean?
Powers of 10 show how many times the number 10 is multiplied by itself. Instead of writing very long numbers with many zeros, we use the exponent to make them short and clean.
Example:
Instead of writing:
\(10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 = 1,000,000,000\)
We write:
10⁹
Here:
10 → base
9 → exponent (power)
It means multiply 10 by 9.
So, \(10⁷ = 10 ×\) itself 7 times.
Meaning of Powers of 10:
Powers of 10 are written like 10ˣ, where x is any integer. If x is Positive. You multiply 10 by itself x times.
Example:
\(10³ = 10 × 10 × 10 = 1000\)
This is the same as writing one followed by three zeros.
If x is Negative
Use the rule:
\(10^{-m} = \frac{1}{10^{m}}\)
Example:
\(10^{-3} = \frac{1}{10^{3}} = \frac{1}{1000} = 0.001\)
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). It is represented by the letter ‘k’ in SI notation, 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:
A negative exponent reduces the number, turning it into a decimal. 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 | Centi (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
(decimal point moves 2 places left).
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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 adding or subtracting, find the common factor (the one with the smaller exponent). Simplify the other number, do the operation, and then multiply the result by the common factor.
For example, find the sum of 106 and 109.
1. Find the smallest exponent: 6 → common factor = 10⁶
2. Factor out 10⁶:
\(10^6 + 10^9 = 10^6 \times (1 + 10^3)\)
3. Simplify inside parentheses:
\(1 + 10^3 = 1 + 1000 = 1001\)
4. Multiply by the common factor:
\(10^6 \times 1001 = 1{,}001{,}000{,}000\)
For example, 105 - 102.
1. Find the smallest exponent: 2 → common factor = 10²
2. Factor out 10²:
\(10^5 - 10^2 = 10^2 \times (10^3 - 1)\)
3. Simplify inside parentheses:
\(10^3 - 1 = 1000 - 1 = 999\)
4. Multiply by the common factor:
\(10^2 \times 999 = 99{,}900\)
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:
\(a^m \times a^n = a^{m+n}\)
For example, \(10^3 \times 10^4
\)
Since the bases are the same (10), we add the exponents.
\(10^3 \times 10^4 = 10^{3+4} = 10^7\)
Now, we find the value of the power of 10.
\(10^7 = 10 × 10 × 10 × 10 × 10 × 10 × 10 = 10,000,000 \)
Thus, \(10^7 = 10,000,000.\)
Let us take another example.
Multiply \(10^{-2} \times 10^5\)
Because both bases are 10, we add the exponents.
\(10^{-2} \times 10^5 = 10^{-2+5} = 10^3\)
Hence, \(10^{-2} \times 10^5 = 10^{-2+5} = 10^3 = 1{,}000\)
Next, we can find the value of 103.
\(10^3 = 10 × 10 × 10 = 1,000 \)
Therefore, \(10^3 = 1,000 \)
Dividing powers of 10:
While dividing numbers, if the bases are the same, we need to subtract the exponents. The rule states that,
\(\frac{a^m}{a^n} = a^{m-n}\)
For example, \(\frac{10^6}{10^4}\).
The bases are the same, so we need to subtract the exponents.
\(\frac{10^6}{10^4} = 10^{6-4} = 10^2\)
\(\frac{10^6}{10^4} = 10^2\)
Next, find the value of 102.
\(10^2 = 10 × 10 = 100\)
Thus, \(10^2 = 100.\)
Let us take another example,
divide \(\frac{10^{-8}}{10^{-14}} = 10^{-8 - (-14)} = 10^{6}\).
\(\frac{10^{-8}}{10^{-14}} = 10^{-8 - (-14)} = 10^6 = 1{,}000{,}000\)
\(10^{-8 + 14} = 10^6\)
\(10^6 = 1{,}000{,}000\)
Expanded Form
A power shows the repeated multiplication of a number. It is written as xⁿ, where n is the exponent and x is an integer. The exponent tells us how many times the base number is multiplied by itself.
Example
If the exponent is 3, it means we multiply the number 10 three times:
\(10³ = 10 × 10 × 10\)
This is repeated multiplication, called the expanded form of the power.
Powers of 10 Chart
A power-of-10 chart helps you understand how the value of 10 changes when the exponent increases or decreases.
- Positive exponents increase the number (adding more zeros).
- Negative exponents are used to make the number smaller (more decimal places).
Example: Positive Power
10⁵ (Expanded Form):
\(10 × 10 × 10 × 10 × 10\)
Decimal Value: \(100000\)
Fraction Form: \(\frac{100000}{1}\)
This means 1 followed by 5 zeros.
Example: Negative Power
10⁻⁵ (Expanded Form):
\(\frac{1}{10 \times 10 \times 10 \times 10 \times 10}\)
Decimal Value: 0.00001
Fraction Form: \(\frac{1}{100000}\)
This means the decimal moves five places to the left.
Here’s a clean and simple Powers of 10 Chart:
| Power of 10 | Expanded Form | Fraction Form | Decimal Value |
| \(10^3\) | \(10 × 10 × 10 \) | 1000 | |
| \(10^2\) | \(10 × 10 \) | 100 | |
| \(10^1\) | 10 | 10 | |
| \(10^0\) | - | 1 | |
| \(10^{-1}\) | \(\frac{1}{10}\) | 0.1 | |
| \(10^{-2}\) | \(\frac{1}{10 \times 10}\) | 0.01 | |
| \(10^{-3}\) | \(\frac{1}{10 \times 10 \times 10}\) | 0.001 | |
| \(10^{-4}\) | \(\frac{1}{10 \times 10 \times 10 \times 10}\) | 0.0001 |
Tips and Tricks for Power of 10
When working with numbers involving powers of 10, we should keep some tips and tricks in mind to solve mathematical problems efficiently and accurately.
- Every number expressed as a power of 10 has two components: the base (10) and the exponent (indicating how many 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”.
- When you multiply by 10ⁿ, the number becomes bigger, and all the digits move to the left.
- When you divide by 10ⁿ, the number becomes smaller, and all the digits move to the right.
- Use real-life situations, such as money, distances, or population numbers, to help children understand how powers of 10 are used to increase or decrease the size of a number.
-
For positive exponents, write 1 followed by the number of zeros equal to the exponent. Example: 10⁵ = 1 followed by 5 zeros → 100000.
-
A powers of 10 chart makes it easy to see the values of both positive and negative exponents, reducing some mistakes. You can even include very large powers like 10¹⁰⁰ to show how huge numbers can become.
-
Visual tools, like the number lines, flashcards, or charts, help children to understand and remember powers of 10 more quickly.
-
Teach children that larger positive exponents mean the larger numbers, while larger negative exponents mean smaller numbers.
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 often assume that a negative exponent produces a negative number, leading to incorrect results. 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, = Hence, 10-3 = 1 / 10³ = 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.
- Space and Astronomy: 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.
- Research: 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.
- Counting Population: 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.
-
Medicine and Biology: In medical laboratories, powers of 10 are used to express the size of microorganisms. For example, the length of a bacterium may be 0.0000001 meter, which is written as 1 × 10⁻⁷ meters in scientific notation.
-
Physics and Engineering: Physicists use powers of 10 to express extremely large or small quantities. For example, the speed of light is about 300,000,000 meters per second, which is written as 3 × 10⁸ m/s in scientific notation for easier understanding.
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 \(10^4 < 10^9\).
Therefore, 104 is the common factor, and we must factor out 104.
\(10^{4} + 10^{9} = 10^{4} \times (1 + 10^{5})\)
After factoring out 104, 1 is remaining from 104, and 105 is remaining from 109.
Next, simplify the expression inside the parentheses.
\(1 + 10^5 = 1 + 10 × 10 × 10 × 10 × 10 = 1 + 100,000 = 100,001 \)
Finally, we can multiply the result by 104.
\(10^{4} \times 100{,}001\)
\(10^{4} = 10 \times 10 \times 10 \times 10 = 10{,}000 \)
\( 10,000 × 100,001 = 1,000,010,000 \)
Therefore, \(10^{4} + 10^{9} = 1{,}000{,}010{,}000\)
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.
\(10^{5} - 10^{2} = 10^{2} \times (10^{3} - 1)\)
After factoring out 102, 1 is remaining from 102 and 103 is remaining from 105.
\(10^{3} - 1 = (10 \times 10 \times 10) - 1 = 1000 - 1 = 999\)
Then, multiply back the common factor 102.
Now, expand \(10^2: 10 × 10 = 100 \)
Next, multiply 100 and 999:
\(100 \times 999 = 99{,}900\)
Therefore, \(10^{5} - 10^{2} = 99{,}900\)
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 = 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:
\(\frac{a^m}{a^n} = a^{m-n}\)
\(\frac{10^8}{10^5} = 10^{8-5} = 10^3\)
Next, find the value of 103.
\(10^3 = 10 \times 10 \times 10 = 1{,}000\)
Therefore, \(\frac{10^8}{10^5} = 10^{8-5} = 10^3 = 1{,}000\)
Problem 5
Divide 10^-9 / 10^-16
10,000,000
Explanation
The rule is:
\(\frac{a^m}{a^n} = a^{m-n}\)
We can apply the rule by subtracting the exponents.
\(\frac{10^{-9}}{10^{-16}} = 10^{-9 - (-16)} = 10^{7}\)
\(10^{-9 + 16} = 10^{7}\)
Now, we can find the value of 107.
\(10^7 = 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 = 10{,}000{,}000 \)
Hence, \(\frac{10^{-9}}{10^{-16}} = 10^{7} = 10{,}000{,}000\)
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 parents help their child practice?
7.How can parents explain what a Power of 10 is to their child?
8.How can parents help their child understand the positive and the negative powers of 10?
9.How can parents explain very large numbers, such as 10 to the power 100?
10.How can parents teach their child to write powers of 10 in decimal form?
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.
