+0

# Very Large Number.

0
399
3

What is the answer?  [9^5,000 / 8^5,000] = ?  Thank you for help.

Aug 20, 2017

#1
+1

IT IS VERY LARGE NUMBER !!!.

5 789 115 800 486 251 073 964 874 278 396 242 570 655 285 960 912 477 271 223 950 874 766 440 360 464 316 528 136 151 126 658 914 148 498 063 833 467 014 352 520 786 050 176 917 397 229 521 120 455 326 332 483 182 545 678 255 558 875 411 689 449 186 373 225 501 637 961 268 294 367 480 668 948 835 153 043 011 813 865 602 874 234 142 765 492.0173384116 2163102620 5846519180 9631915396 2812014182 6770147184 ............

Aug 20, 2017
#2
+2

Well,you can use laws of indices to write this this as

(9/8)^5000   = (1.125)^5000  which is still a gigantic number.

If you now use the binomial theorem  and look at the first couple of terms you can see how quickly its series grows

We have ( 1 + x) ^n     = 1 +nx +[n(n-1)x^2]/2!    + [n(n-1)(n-2)x^3]/3! ...........

I'll just evaluate the first couple of terms to get

1  + 5000(0.125)  +[5000(4999)(0.125)^2]/2!       ......

= 1 + 625 + 195,273 ....

so by the time we only even get to the third term we are  in the hundreds of thousands. The 5000th term?

Input (1.125)^5000 and see if you can make sense of the number!

Aug 20, 2017
#3
+1

9^5,000 / 8^5,000

$$let\\ y=\frac{9^{5,000 }}{8^{5,000}}\\ y=(\frac{9}{8})^{5000}\\ logy=log(\frac{9}{8})^{5000}\\ logy=5000log(\frac{9}{8})\\$$

5000*0.0511525224473813 = 255.7626122369065

$$logy= 255.7626122369065\\ 10^{logy}= 10^{255.7626122369065}\\ y= 10^{255}*10^{0.7626122369065}\\$$

10^0.7626122369065 = 5.7891158004869876

$$y=5.7891158004869876\times 10^{255}\\~\\ 9^{5,000}/8^{5,000}=5.7891158004869876\times 10^{255}$$

.
Aug 21, 2017