   SEARCH HOME Math Central Quandaries & Queries  Question from Kent, a student: How to write 8 365 005 106 in expanded and exponential form? May I request you also to explane how does it solve. Thank you. Hi Kent.

"Exponential form" simply means a numeric form involving exponents.

One way to write such a number is by recognizing that each position represents a power (exponent) of 10. So you can first break it up into separate pieces.

I will use my own number 5 625 702:
5 625 702 = 5 000 000 + 600 000 + 20 000 + 5 000 + 700 + 2

Each of these is a multiple of the number with a 1 in front:
5 625 702
= 5 x 1 000 000 + 6 x 100 000 + 2 x 10 000 + 5 x 1 000 + 7 x 100 + 2

and each of these is a power of ten, so you can write the exponent which
is the same as the number of zeros:

5 625 702 = 5 x 106 + 6 x 105 + 2 x 104 + 5 x 103 + 7 x 102 + 2 x 100

That last one may not really be necessary. 10^0 is equal to 1, so really we are just saying 2 x 1.

That is one expanded exponential form.

Another way to represent a number using exponents is to write it as a product of prime numbers. Any positive integer above 1 is the product of a unique set of prime numbers.

I know that the number 26 460 is 2 x 2 x 3 x 3 x 3 x 5 x 7 x 7.

So written using exponential form as a product of primes, I write

26460 = 22 x 33 x 5 x 72

The exponents reflect how many times that prime number appears as a factor.

There is also an exponential form called "scientific notation" which is used extensively in science. That is simply a matter of counting how many digits you need to move the decimal place in order to have a number between 1 and 9.999...

So for 5 625 702, I would move the decimal place to 5.625 702, which is 6 places leftwards from where it was. Thus, I say that

5 625 702 in scientific notation is writting 5.625 702 x 106.

I hope this gives you a good idea of how to write things in exponential forms.

Cheers,
Stephen La Rocque.     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.