Power of 10

Dhristi JEE 2022-24

Introduction to Power of 10

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.


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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

101 = 10

100 = 1

102 = 100

10-1 = 0.1

103 = 1000

10-2 = 0.01

104 = 10,000

10-3 = 0.001

105 = 100,000

(one hundred thousand)

10-4 = 0.0001

(one ten thousandth)

106 = 1,000,000

(one million)

10-5 = 0.00001

(one hundred thousandth)

107 = 10,000,000

(ten million)

10-6 = 0.000001

(one millionth)

108 = 100,000,000

(one hundred million)

10-7 = 0.0000001

(one ten millionth)

109 = 1,000,000,000

(one billion)

10-8 = 0.00000001

(one hundred millionth)

1010 (10 to the tenth power) = 10,000,000,000

(ten billion)

10-9 = 0.000000001

(one billionth)


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

1. What is Meant by Logarithm?

Logarithm, the power or exponent to which a base should be raised to produce a given number. Written mathematically, x is the logarithm of n to the base b if bx = n, where case one writes x = logb n. For example, 24 = 16; thus, 4 is the logarithm of 16 to base 2, or 4 = log2 16. Similarly, since 103 = 1000, then 3 = log10 1000. Logarithms of the second-mentioned sort (that is, logarithms with base 10) are known as common, or Briggsian, logarithms and are simply written log n.

2. What is the importance of ‘logs’ in powers?

Are you assuming if there's an opposite to powers of 10? Something like how multiplication is the opposite of division or addition is the opposite of subtraction. It appears that there is such a thing called: ``logs'' or “logarithms”. Logs initially emerged to be important for multiplying and dividing and were used all the time in performing arithmetic operations with the help of slide rules. Now that calculators are commonplace, the use of logarithms for fundamental calculations is steadily disappearing, but it can still be worthwhile. And while employing logarithms for simple arithmetic calculations is not so common, there are other uses for logarithms in many areas of science.

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