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Algebra Level 2

$\LARGE 3^{2^{85}} = \, ?$

Note: Exponent towers are evaluated as $\large a ^ {b^c} = a^{ \left( b^ c \right) }$ .

\large 9^{85}

\large 3^{4^{84}}

\large 9^{2^{42}}

\large 9^{4^{42}}

4 solutions

Muhammad Rasel Parvej
Mar 14, 2017

$\Large 3^{2^{85}} = 3^{2\times 2^{84}} = \left( 3^2\right)^{2^{84}}= 9^{2^{84}} = 9^{4^{42}}.$

Moderator note:

This problem shows that it can be tricky to eyeball the equivalence of exponent towers. When in doubt, simplify it to a form that you're familiar with, by taking common bases.

Yash Jain
Mar 20, 2017

As According to Exponent Towers ,

$\large 3^{(2^{85})} = 3^{2^1 \times 2^{84}}= (3^2)^{2^{84}} = 9^{2^{84}}$

We have, $\large 2^{84} = 4^{42}$

$\large \therefore \quad 9^{2^{84}} = \boxed{9^{4^{42}}}$

are you a math prodigy "?""

Nuwelin Dagne - 4 years, 2 months ago

I still think I was right. It depends on the convention you follow.

Rodney Hampson - 4 years, 2 months ago

9^4^42 is the same thing as 9^2^84 because it is like simply putting parentheses in a sea of multiplication. (2x2)(2x2)(2x2)(2x2), etc. The number of twos multiplied together, regardless of parentheses, is 84. 9^2^84 = 9^(2^84) = (3x3)^(2^84) = (3x3)(3x3)(3x3)(3x3) etc. The number of 3s multiplied is 2(2^84). This number is also equal to 2^85. Therefore 9^4^42 = 9^2^84 = 3^2^85.

Jessie Mondragon - 2 years, 8 months ago

Joy Patel
Mar 30, 2017

3^2^85=3^2×2×2...85 times = ((3^2)^2)^2)....85 times= (9^2^84)=9^4^42 { as 2×2×2×2.... 84 times= (2×2)×(2×2)...84 times = 4^84}

Cheers!

Let 3^(2^(85)) = (3^2)^x

3^(2^(85)) = 3^(2x)

Then 2^85 = 2x

2^84 = x

2^(2*42) = x

(2^2)^42 = x

4^42 = x

So, 3^(2^(85)) = 9^(4^(42)).

Dennis Rodman - 2 years, 2 months ago

Oon Han
Jul 7, 2019

Top to bottom! $\begin{aligned} 3^{2^{85}} &= 3^{2 \times 2^{84}} \\ &= {(3^2)}^{2^{2 \times 42}} \\ &= 9^{{(2^2)}^{42}} \\ &= \boxed{9^{4^{42}}} \end{aligned}$

Therefore, the answer is $9^{4^{42}}$ .

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