Dina's Dilemma on Tape
Dina was asked by her teacher to evaluate 9^{99}. Just before she plunged in, however, she decided to approximate the expected size of her final answer. She assumed that she could write 5 digits per inch along an adding machine tape. How long of a strip will she need?
And if you want to explore the situation further, repeat the problem for the following numbers: 9, 9^{9}, 9^{999}, etc.
Source: Adapted from Oregon Mathematics Teacher, Sept. 2000, p. 44
Hint: Remember that 9^{99} means [9^{(99)}].
Solution Commentary: Let x = 9^{99} = 9^{387420489}
log(x) = 387420489 log(9)= 369693099.6...
Thus, x has about 10^{369693099.6} digits, or 369,693,100 digits
Which means (at 5 digits/inch), Dina will need 73,938,619.8 inches of tape, or 6,161,551.65 feet of tape, or about 1,167 miles of tape!
