   SEARCH HOME Math Central Quandaries & Queries  Question from dylan, a student: how do you write 20736 in exponential form .same for 1728 and 50625. is there a formula to figure out how to express large know numbers in exponential form. Hi Dylan,

There may be many ways to write a number in exponential form. For examples

16 = 42
16 = 4 × 22
16 = 24.

Each of these is in exponential form since it uses exponents (powers). I expect that you want to express the numbers you have as powers of primes so for 16 you wound get 16 = 24.

There is no easy way to write a number as a product of primes. You can start by dividing by small primes and see where that gets you. This is what Stephen did in his response to a similar question. With your numbers I know where to start. 20736 and 1728 are even so you can divide by 2 and proceed. 50625 has 5 as its last digit so it is divisible by 5. Hence you can divide by 5 and proceed. Here I have used the facts

if the last digit of a number is even then the number is even, and
if the last digit of a number is 5 or 0 the number is divisible by 5.

There are two other fact that you might find useful

if the sum of the digits of a number is divisible by 9 the number is divisible by 9, and
if the sum of the digits of a number is divisible by 3 the number is divisible by 3.

Hence for your number 1728, 1 + 7 + 2 + 8 = 18 which is divisible by 9 so 1728 is divisible by 9, thus

1728/9 = 192.

1 + 9 + 2 = 12 which is divisible by 3 so 192 is divisible by 3, and thus

192/3 = 64.

But 64 = 82 and 8 = 23 so I get

1728 = 9 × 192 = 32 × 3 × 64 = 33 × 23 × 23 = 26 × 33.

Now try the other two numbers.

Penny     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.