Powers of 10
The powers of 10 refer to the numbers in which the base is 10 and the exponent is an integer. For example, 102, 103, 106 show the different powers of 10. This can be understood with the concept that when 10 is multiplied a specific number of times, then it can be expressed in the form of exponents and those are called the powers of 10. Let us learn more about the powers of 10 in this page.
| 1. | What does Powers of 10 mean? |
| 2. | 10 to the Power of 2 |
| 3. | 10 to the Power of 3 |
| 4. | 10 to the Power of 1 |
| 5. | Powers of 10 Chart |
| 6. | FAQs on Power of 10 |
What does Powers of 10 mean?
The powers of 10 means when 10 is multiplied a certain number of times, the product can be expressed using exponents. These numbers which are written as exponents are the powers of 10. If we multiply 10 a couple of times it becomes difficult to write the number as in this case, 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 = 1000000000. Now, if we need to multiply 10 thirty times, it would be even more difficult to write the product with so many zeros. Therefore, exponents help to express this easily and this value (10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 = 1000000000) can be expressed as 109. Here, 10 is the base and 9 is the power and this is read as 10 to the ninth power. Now, let us try to understand it the other way round. For example, 10 to the 7th power means 107. This means that we need to multiply 10 seven times, that is, 107 = 10 × 10 × 10 × 10 × 10 × 10 × 10
This can be explained in another way.
The powers of 10 are of the form 10x, where x is an integer. 10x is read as '10 to the power of x'. If x is positive, we simplify 10x by multiplying 10 by itself x times. For example, 103 = 10 × 10 ×10 (3 times) = 1000. If x is negative, then we apply the property of exponents, a-m = 1/am and then we apply the same logic as explained earlier. For example 10-3 = 1/103 = (1/10)3 = 1/10 × 1/10 × 1/10 = 1/1000 = 0.001. By using these two examples, we can conclude two things that are very useful to calculate the powers of 10.
- When the power is positive, 10x = '1 followed by x number of zeros'.
For example, 106 = 1,000,000. Here, there are 6 zeros placed after 1 because the power of 10 is 6. - When the power is negative, 10-x = '0 point followed by (x -1) number of zeros followed by 1".
For example, 10-6 = 0.000001. Here, we placed 5 zeros after the decimal point (followed by 1) as the power was a negative 6 and 6 - 1 = 5.
10 to the Power of 2
10 to the power of 2 is also called the second power of ten. This is written as 102 and this means that 10 is multiplied two times. In other words, 10 × 10 = 102. Here 10 is the base and 2 is the exponent. This can be further evaluated as 102 = 100.
10 to the Power of 3
10 to the power of 3 is called the third power of ten and is written as 103. This means, 10 × 10 × 10 = 103. In this expression, 10 to the third power, 10 is the base and 3 is its power or exponent. This can also be evaluated as 103 = 1000.
10 to the Power of 1
10 to the power of 1 means the first power of ten which is 101. We know that any number to the power of 1 means it is the number itself. So here, 101 = 10.
Powers of 10 Chart
The powers of 10 chart shows that the different powers of 10 have different values. For example, if we write 105 in the expanded form, it will be 105 = 10 × 10 × 10 × 10 × 10. Now, the value of 105 in the decimal form will be 100000. And if we write it in the form of a fraction it will be 100000/1. Similarly, if we write 10-5 in the expanded form, it will be 10-5 = 1/(10 × 10 × 10 × 10 × 10). Now, the value of 10-5 in the decimal form will be 0.00001. And if we write it in the form of a fraction it will be 1/100000. The following table shows the powers of 10 chart which includes positive powers and negative powers.
Positive Powers of 10
The powers of 10 have some specific names (though not all powers) for some specific powers. For example, 106 (10 to the power of 6) is known as a 'million' and the SI prefix of 10 power 6 is 'giga' which is represented by the SI symbol G. Similarly, we have some specific names for some positive powers of 10 which are given in the following table.
| Positive Powers of 10 | Name | Prefix (Symbol) |
|---|---|---|
| 101 = 10 | Ten | Deca (D) |
| 102 = 100 | Hundred | Hecto (H) |
| 103 = 1000 | Thousand | Kilo (K) |
| 106 = 1,000,000 | Million | Mega (M) |
| 109 = 1,000,000,000 | Billion | Giga (G) |
| 1012 = 1,000,000,000,000 | Trillion | Tera (T) |
| 1015 = 1,000,000,000,000,000 | Quadrillion | Peta (P) |
| 1018 = 1,000,000,000,000,000,000 | Quintillion | Exa (E) |
| 1021 = 1,000,000,000,000,000,000,000 | Sextillion | Zetta (Z) |
| 1024 = 1,000,000,000,000,000,000,000,000 | Septillion | Yotta (Y) |
Negative Powers of 10
The negative powers of 10 are expressed in a different way. We know that a negative power (negative exponent) is defined as the multiplicative inverse of the base. This means that we write the reciprocal of the number and then solve it like positive exponents. For example, (4/5)-2 can be written as (5/4)2. Similarly, a negative power of 10, like 10-3, can be written as 1/103, or, 1/(10 × 10 × 10) = 1/1000 = 0.001
Just like how we have some peculiar names for positive powers of 10, we have some names for some negative powers of 10 as well. A few of them are given in the following table.
| 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) |
| 10-15 = 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) |
2 to the Power of 10
It should be noted that 2 to the power of 10 is not the same as 10 to the power of 2. 2 to the power of 10 means a number in which 2 is the base and 10 is the exponent. This is written as 210 and this means 2 is multiplied ten times, that is, 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1024.
Calculating Powers of 10
In order to calculate the sum, difference, product, and quotient of powers of 10, we can first find the values of powers of 10 and then do the respective operation. For example, 103/102 = 1000/100 = 10. But sometimes, this procedure is difficult if the exponent is very large. In such cases, the following procedures would help.
Adding and Subtracting Powers of 10
To add and subtract powers of 10, we take the minimum power of 10 as the common factor and then simplify the rest. For example,
- 105 + 108 = 105 (1 + 103) = 105 (1 + 1000) = 105 (1001) = 100,100,000
Multiplying Powers of 10
To multiply the powers of 10, we apply an exponent rule that says am × an = am + n. This rule says that we need to add the exponents when the bases are the same. Hence, this rule can be applied to multiply two or more powers of 10. Here are some examples.
- 105 × 108 = 105 + 8 = 1013
- 10-3 × 106 = 10-3 + 6 = 103
Dividing Powers of 10
There is a rule of exponents, am / an = am - n. We use this rule to divide the powers of 10. This rule says that we need to subtract the powers when the bases are the same. Here are a few examples.
- 1017 / 1015 = 1017 - 15 = 102 = 100
- 10-6 / 10-12 = 10-6 + 12 = 106
Important Tips on Powers of 10
- Powers of 10 refer to numbers like 105, or 106, where 10 is the base and 5 and 6 are its powers.
- 2 to the power of 10 means a number in which 2 is the base and 10 is the exponent, that is, 210.
- Just like 2 to the power of 10 means 210, other phrases like 3 to the power of 10 means 310, 4 to the power of 10 means 410. These should not be confused with the powers of 10 that we have studied on this page.
☛ Related Topics
FAQs on Power of 10
What are the Powers of 10 in Math?
Powers of 10 refer to the numbers in which 10 is the base and any integer is the exponent. For example, 103, 106, 10-7 are a few examples of the powers of 10.
How much is 10 to the Power of 10?
10 to the power of 10 means an expression in which 10 is the base and 10 is the exponent. This can be expressed as 1010 and this means 10 is multiplied 10 times, that is, 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 which is equal to 10000000000.
How to Convert 0.00001 to the power of 10?
In order to convert 0.00001 to the power of 10, first, we need to convert this decimal into its fraction form. This will make it 1/100000. Now, this fraction can be written in an exponential form which will be 1/105. This can be further expressed as a negative exponent,10-5
How to Convert a Number to the Power of 10?
In order to convert a number to the power of 10, we write it in the scientific notation. For example, the number 5040000000000000 is a little difficult to write and it would be easier if we write it in the standard form where we use the powers of 10. So this is expressed as 5.04 × 1015. Let us see how to write this number in the standard exponential form, using the following steps:
- Step 1: Count the number of trailing zeros in the given number. In the given number, 5040000000000000, the number of trailing zeros are 13.
- Step 2: Use the beginning part of the given number and write the digits from the left till the last non-zero digit, followed by a 10 raised to a power that is equal to the number of trailing zeros. This means 504 and 1013
- Step 3: Place a decimal point after the first digit from the left side and add the number of decimal places that are created, to the power of 10 which is written. Here, we will place a decimal point after 5, and it will become 5.04. Since there are 2 decimal places created in 5.04, we will add 2 to the existing power of 10. The existing power of 10 was 13 because there were 13 trailing zeros, but now it will become 15. This will make it 5.04 × 1015. Therefore, 5040000000000000 can be written as 5.04 × 1015
How to write 100 as a Power of 10?
In order to write 100 as a power of 10, we will first count the number of zeros in 100, which is two. This means 100 = 10 × 10. Therefore, 100 as a power of 10 can be written as 102
What is the Second Power of 10?
The second power of 10 can be written as 102. This is also known as 10 to the power of 2 and is equal to 100 because 102 = 10 × 10 = 100.
What is the First Power of 10?
The first power of 10 is written as 101. This is also read as 10 to the power of 1 and we know that any number to the power of 1 is the number itself, so 101 = 10.
How to Multiply Decimals by Powers of 10?
In order to multiply decimals by powers of 10, we need to remember a simple rule. We express the product in such a way that we write the given decimal number and move the decimal point to the right according to the number given as the exponent of 10. If the exponent of 10 is 3, we will write the given number and move the decimal 3 places to the right to get the answer easily. For example, if we need to multiply 46.3 × 104, we can see that the exponent of 10 is 4, so we will move the decimal point 4 places to the right. This means, 46.3 × 104 = 463000.
How much is 2 to the Power of 10?
2 to the power of 10 means an expression where 2 is the base and 10 is the exponent. This can be expressed as 210 and this means that 2 is multiplied ten times, that is, 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1024.
How to Write a Given Number as a Power of 10?
- To express a given number (>1) as a power of 10, just write it as 10n (n being positive) where 'n' is the number of zeros after 1 in the given number. For example, 10000 = 104.
- To express a given number (<1) as a power of 10, just count the number of zeros after '0 point' and before '1', add 1 to the result, and then put that number along with the negative sign as the exponent of 10. For example, 0.001 = 10-3 (as there are 2 zeros after 0 point and before 1 in 0.001).
What is the Difference Between 2 to the Power of 10 and 10 to the Power of 2?
2 to the power of 10 = 210 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1,024 whereas 10 to the power of 2 is 102 = 10 × 10 = 100. Thus,
- 2 to the power of 10 = 1,024
- 10 to the power of 2 = 100
How to Find the Powers of 10?
To find the powers of 10, we use the following shortcuts depending upon whether the exponent is positive or negative.
- If the exponent is positive, then 10n = '1 followed by 'n' zeros'. For example, 104 = 10000.
- If the exponent is negative, then 10n = '0 point followed by (n - 1) zeros followed by 1'. For example, 10-4 = 0.0001.