How to Use Powers of 10 for Quick Calculations
The power of ten, in math, any of the integer (whole-valued) exponents of the number 10. A power of 10 is as many number 10s as designated by the exponent multiplied together. Therefore, shown in long-form, a power of ten is the number 1 subsequent to n zeros, in which ‘n’ is the exponent and is greater than 0; for example, 106 is mathematically written as 1,000,000. When n is less than 0, the power of 10 is the number 1 n placed followed by the decimal point; for example, 10−3 is written 0.001.
Power of 10 to Zero
When n equals to 0, the power of 10 will be 1; i.e., 100 = 1. Refer to the below table for expressions in positives and negative powers of 10.
Powers of 10
Multiplying and Dividing Powers of 10
Maybe you're not sure that it's useful to be able to write numbers in powers of ten math or mega 10 powers. Well, powers of ten math are quite helpful when performing math calculations, as well. Let's say if we ask you ''what are 10 times 1,000?'' Hardly anything- it's just 10,000. But what if we ask ''what will be one trillion times one quadrillion?''
It concludes that the multiplication of really large numbers is easy with powers of ten. All you need to do is to add up the exponents, and you're sorted. Let's consider the example we just quoted above. What is one trillion times one quadrillion? First, one trillion is 1012, and one quadrillion is 1015. So the correct answer is 1027, which is a huge number. Now, you can see that almost immediately, without requiring a calculator, we did it shorthand.
Power of Ten Prefixes
When a numerical digit represents a quantity instead of a count, SI prefixes can be used - therefore "femtosecond", not "one quadrillionth of a second'' - although most frequently powers of 10 are used rather than some of the very low and very high prefixes. In certain cases, specialized units are taken help of, such as the light-year particle physicists barn or the astronomer's parsec.
Nonetheless, large numbers carry an intellectual intrigue and are of mathematical interest, and assigning them names is one of the ways in which most of us try to conceptualize and understand them.
Solved Examples
Solve the following expressions:
Log (106) = 6
Log (1027) = 27
Log (10365.2748) = 365.2748
Log (10-5) = -5
Log (x) -5 → x = 105
Log (x) = 6.789 → x = 106.789
Log (x) = -2.23 → x = 10-2.23
Fun Facts
The power of 10 is easy to remember since we use base 10 of a number system.
For 10n having ‘n’ a positive integer, just write "1" with n zeros after it. For negative powers 10−n, write “0" followed by n−1 zeros, and then a 1. The powers of 10 are extensively used in scientific notation.
Power of 10
Let's take a number 10. We could take two 10s and multiply them together, which means 10 times 10, which you know is equal to 100. We could also take three 10s and multiply them together, 10 times 10 times 10 which is equal to one thousand. And we could do this with any number of 10s. But at some point, if we do this with enough 10s, it will get pretty hard for us to write. So let's give an example. Let's say I were to do this with ten 10s, so if I were to go 10 times 10 just like this : 10×10×10×10×10×10×10×10×10. This is going to be equal to even the number that is equal to, is going to be quite hard to write. It is going to be one that is followed by ten zeros. This will be 10 billion. And it's already getting hard to write. And imagine if you have thirty 10s that we were multiplying together, it will be very much difficult to calculate or write down like this.
The mathematicians have come up with some notations and some ideas to be able to write things like this, a little bit more elegantly. So the way they do this is through something which is known as exponents. And so 10 times 10, we can rewrite as being equal to, if I have two 10s and I'm multiplying them together, I could write this as 10 to the second power. That's how we can pronounce it as 10 to the power 2. It looks quite fancy but all that means is let's take two 10s and multiply them together and we are going to get one hundred. In this, the two would be called as exponents and the 10 would be the base. So eventually, 10 to the second power of 10 times 10 is equal to hundred.
So how would you write 10 times 10 times 10 or 10000 ? How are you going to write that by using exponents ? We are taking three numbers of 10s and multiplying them together, this would be 10 to the third power. Here, ten is the base and three is the exponent. We would read this as 10 to the third power. If you will ever see 10 to the third power, that means we can multiply 10 times 10 times 10 which is the same thing as one thousand. So this is another way of writing 1000.
FAQs on Power of 10 Explained: Meaning, Rules & Applications
1. What does the term 'power of 10' mean in mathematics?
In mathematics, the 'power of 10' refers to a number in the form 10ⁿ, where 10 is the base and 'n' is the exponent. The exponent indicates how many times the number 10 is multiplied by itself. For example, 10⁴ means 10 × 10 × 10 × 10, which equals 10,000. It is a fundamental concept for writing very large or very small numbers compactly.
2. How do you calculate a number raised to the power of 10?
To calculate a positive integer power of 10, you simply write the digit '1' followed by the number of zeros indicated by the exponent. For instance, to calculate 10⁶, you write a '1' followed by six zeros, which gives you 1,000,000 (one million).
3. Can you provide some common examples of powers of 10?
Powers of 10 are used to represent numbers of all sizes. Here are some common examples:
- 10³ = 1,000 (One Thousand)
- 10⁶ = 1,000,000 (One Million)
- 10⁹ = 1,000,000,000 (One Billion)
- 10⁰ = 1 (One)
- 10⁻¹ = 0.1 (One-Tenth)
- 10⁻² = 0.01 (One-Hundredth)
4. What is the importance of using powers of 10 in science?
The primary importance of powers of 10 in science is their use in scientific notation. This system allows scientists to express extremely large numbers (like the distance to a star in kilometres) or extremely small numbers (like the diameter of an atom) in a standard, concise format. This simplifies complex calculations and reduces the risk of errors when handling numbers with many digits.
5. What happens when 10 is raised to a negative power, like 10⁻²?
A negative exponent indicates a reciprocal, meaning division. When 10 is raised to a negative power, it represents a number smaller than 1. Specifically, 10⁻ⁿ = 1 / 10ⁿ. Therefore, 10⁻² is equivalent to 1 / 10², which equals 1/100 or the decimal 0.01. This is essential for representing fractions and decimals in the base-10 system.
6. How is a large number like one billion written using the power of 10?
To write a large number like one billion as a power of 10, you count the number of zeros after the '1'. A billion is written as 1,000,000,000, which has nine zeros. Therefore, one billion can be expressed concisely as 10⁹.
7. How are powers of 10 related to the decimal number system we use every day?
The decimal system is a base-10 system, which means it is fundamentally built on powers of 10. Each digit's position in a number represents a specific power of 10. For example, in the number 452, the '4' is in the hundreds place (10²), the '5' is in the tens place (10¹), and the '2' is in the ones place (10⁰). This place value system is a direct application of powers of 10.
8. Why is any number raised to the power of zero equal to 1, including 10⁰?
This can be understood by observing the pattern of exponents. Consider the sequence: 10³ = 1000, 10² = 100, 10¹ = 10. Each time the exponent decreases by one, the result is divided by 10. Following this pattern, to get from 10¹ to 10⁰, we would divide the result (10) by 10, which gives 1. This is why 10⁰ = 1, a consistent rule in the laws of exponents.
9. Is there a special name for 10 raised to the power of 100?
Yes, the number 10¹⁰⁰ (a 1 followed by 100 zeros) is known by the special name googol. It's a famous example used in mathematics to help conceptualise the scale of very large finite numbers and distinguish them from the concept of infinity.
