# Which is bigger? 99^101 or 101^99?

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• 99^101 because its being raised by a higher exponent.

• Anonymous
5 months ago

99¹⁰¹ > 101⁹⁹

Source(s): Been there
• You would think that 99^101 < 101^99 just because you have a greater base. But the exponents in this case are the bigger telltale. First I'd use simpler powers of the bases. I'll go up to 5 for each of them.

101^1 -- self explanatory.

101^2 -- 10,201 (5 digits).

101^3 -- 1,030,301 (7 digits).

101^4 -- 104,060,401 (9 digits).

101^5 -- 10,510,100,501 (11 digits).

99^1 -- self explanatory.

99^2 -- 9,801 (4 digits).

99^3 -- 970,299 (6 digits).

99^4 -- 96,059,601 (8 digits).

99^5 -- 9,509,900,499 (10 digits).

See the patterns? 101^x will have 2x + 1 digits if x is a whole number at least 1, so 101^99 will have 2(99) + 1 = 199 digits. 99^x will have 2x digits with the same restriction for x, so 99^99 will only have 2(99) = 198 digits and won't surpass 101^99. But 99^100 will -- it would have 200 digits, and 99^101 would have 202 digits. The verdict:

99^101 > 101^99.

• Johnathan
Lv 7
5 months agoReport

Ok, thanks. 😁

• Try this 99^101=99^100 *99. 101^99=101^100/101. Divide one by the other

(99/101)^100 (101/99). The first term is extremely small. Therefore 101^99 is bigger the number.

• You can guess the bigger power will win.

The bases are almost the same number (basically 100±1%)

Whereas the powers being different by 2 actually results in factor change of approx 100^2 = 10,000

• L := log_10.

99^101 vs 101^99

L(99^101) vs L(101^99)

101L99 vs 99L101

L(99)/99 vs L(101)/101

Consider the function L(x)/x. This is a decreasing function (for x > 3) Hence,

L(99)/99 > L(101)/101. Going back up the steps,

99^101 > 101^99

• 99^101 is bigger

• 99^101/101^99 =

1353.12728018

• METHOD #1

log(99^101) = 101 log(99) = 201.559

log(101^99) = 99 log(101) = 198.42

If log(a) > log(b) then a > b.

METHOD #2

99^101 / 101^99 = (99/101)^99 * 99^2

= 1353...

If a/b > 1 then a > b.

METHOD #3 - WITHOUT A CALCULATOR

(combine methods 1 and 2)

log(99^101) / log(101^99)

= (101 log(99)) / (99 log(101))

= (101/99) * (log(99) / log(101))

= (101/99) * (log(99) / log(101))

The log curve is much flatter than linear, so (101/99) > (log(101) / log(99)).

(101/99) * (log(99) / log(101)) > 1

• 99^101 > 101^99

99^101

= 3.6237201786049720... × 10^201

101^99

=

2.678033494476758508... × 10^198