Very large exponent

**cerpintaxt33** · June 12th 2010, 11:29 AM

I need help defining the following very large exponent:

10,000 ^ 13,766,400

No doubt, this number is astronomical...but surely (hopefully) it can be expressed with scientific notation. The calculation has maxed out my TI-83 as well as any online calculator designed to handle large calculations. I do not consider myself to be mathematically inclined, so I have no way of knowing whether or not I am asking for something absurd. I appreciate any help in this matter.

**Plato** · June 12th 2010, 11:36 AM

Originally Posted by cerpintaxt33

I need help defining the following very large exponent:

10,000 ^ 13,766,400

Here is an example: $(10,000)^{321}=(10^4)^{321}=(10)^{1284}$

**CaptainBlack** · June 12th 2010, 11:36 AM

Originally Posted by cerpintaxt33

I need help defining the following very large exponent:

10,000 ^ 13,766,400

No doubt, this number is astronomical...but surely (hopefully) it can be expressed with scientific notation. The calculation has maxed out my TI-83 as well as any online calculator designed to handle large calculations. I do not consider myself to be mathematically inclined, so I have no way of knowing whether or not I am asking for something absurd. I appreciate any help in this matter.

In scientific notation it is: $1 \times 10^{13,766,404}$

You should not need a calculator for this just the definition of scientific notation and the laws of exponents.

(most calculators and computer languages are limited to decimal exponents of 3 or less digits, some not being able to handle even 3 digits)

CB

**mfetch22** · June 12th 2010, 11:38 AM

Originally Posted by cerpintaxt33

I need help defining the following very large exponent:

10,000 ^ 13,766,400

No doubt, this number is astronomical...but surely (hopefully) it can be expressed with scientific notation. The calculation has maxed out my TI-83 as well as any online calculator designed to handle large calculations. I do not consider myself to be mathematically inclined, so I have no way of knowing whether or not I am asking for something absurd. I appreciate any help in this matter.

Note that $10^4 = 10,000$

So, since sicientfic notation is expressed in base 10, and the number in front of the 10 in this case is 1 we have:

$(10,000)^{13,766,400} = (10^4)^{13,766,400}$

And an exponant to an exponant is simply the exponants multiplied together so we have:

$(10,000)^{13,766,400} = 10^{4*13,766,400} = 10^{55,065,600}$

And since its multiplied by 1 we have:

$1$ X $10^{55,065,600}$

**cerpintaxt33** · June 12th 2010, 01:24 PM

Thanks, I appreciate it. Any idea how many digits this number would be?

**mfetch22** · June 12th 2010, 01:36 PM

Originally Posted by cerpintaxt33

Thanks, I appreciate it. Any idea how many digits this number would be?

Well since $10^n$ adds $n-1$ zeros to $10$ (and thus $n-1$ digits ), and becuause 10 already adds two digits, I beleive that this number would have $55,065,601$ digits

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