Large Exponents

Algebra Level 1

We know that

$\huge 5^{4^{3^{2^1}}} > 1^{2^{3^{4^5}}} .$

Which is larger,

$\huge 5^{4^{3^{2}}} \text{ or } {2^{3^{4^5}}}?$

5^{4^{3^{2}}}

{2^{3^{4^5}}}

They are equal

3 solutions

Zee Ell
Jan 26, 2017

$5^ { 4 ^ {3 ^ 2} } < 8 ^ { 4 ^ {3 ^ 2} } = (2^3)^ { 4 ^ {3 ^ 2} } = 2^ { 3 × 4 ^ {3 ^ 2} } = 2^ { 3×4^9 } < 2^ { 4×4^9 } = 2^ { 4^{10} } = 2^ { 2^{20} }$

$2^ { 3 ^ {4 ^ 5} } > 2^ { 2 ^ {4 ^ 5} } = 2^ { 2 ^ {1024} }$

Hence:

$5^ { 4 ^ {3 ^ 2} } < 2^ { 2^{20} } < 2^ { 2 ^ {1024} } < 2^ { 3 ^ {4 ^ 5} }$

Therefore, our answer should be:

$\boxed { 2^ { 3 ^ {4 ^ 5}} }$

That's a nice way to compare them. Converting to powers of 2 makes the calculations much easier.

Chung Kevin - 4 years, 4 months ago

ERROR! Stacked powers are evaluated left to right, so your expression 4^3^2 should be evaluated as (4^3)^2 or 4^6 NOT 4^(3^2) = 4^9 as you have a^b^c = a^(b*c)

Fay Rowland - 4 years, 3 months ago

Unfortunately, you made a mistake (and that's a very common mistake to make with exponent towers). See how are exponent towers evaluated?

The way that I remember it, is that if a^b^c = a^(bc), then we would not need to introduce such complicated notation. So, there has to be a reason why it's written as a tower and not as just a single term, so a^b^c = a^(b^c)

Chung Kevin - 4 years, 3 months ago

A Former Brilliant Member
Feb 2, 2017

Note: The following solution is provided by an online friend keith291 from Taiwan's forum telnet://ptt.cc or https://www.ptt.cc/bbs/Math/

LHS = 5^(4^(3^2)) = 5^(4^9) = 5^((2^2)^9) = 5^(2^(2*9)) = 5^(2^18)

RHS = 2^(3^(4^5)) = 2^(3^1024) = 2^(3*3^1023) = (2^3)^(3^1023) = 8^(3^1023)

5 < 8 and 2^18 < 3^1023 $\implies$ LHS < RHS

Edit: THE FOLLOWING SOLUTION IS WRONG.

LHS = ((5^4)^3)^2 = 5^(4*3*2) = (5^2)^(4*3) = 25^12

RHS = ((2^3)^4)^5 = 2^(3*4* 5) = (2^5)^(4*3) = 32^12

obviously LHS < RHS

That's not how how exponent towers are evaluated .

Chung Kevin - 4 years, 4 months ago

You are right! I am wrong and have not figure out an elegant solution yet. Should I delete my wrong solution immediately, or may we discuss how to modify the original problem so that I will choose a wrong answer if I think in this wrong way of evaluating a " nested exponentials "?

A Former Brilliant Member - 4 years, 4 months ago

I edited and supplemented a "concise" correct solution contributed by an online friend keith291 .

A Former Brilliant Member - 4 years, 4 months ago

Looks good now :)

You should invite your friend to Brilliant.

Chung Kevin - 4 years, 4 months ago

Osman Ates
Feb 1, 2017

5^4^3^2=5^2^4^3=25^4^3=25^{12} 2^3^4^5=2^5^4^3=32^4^3=32^{12} 32^{12}>25^{12}

2^3^4^5>5^4^3^2

That's not how how exponent towers are evaluated .

Chung Kevin - 4 years, 4 months ago

You cannot multiply the exponents in an exponent tower. $a^{b^{c^{d}}} \neq \left(\left(a^b\right)^c\right)^d = a^{bcd}$ . You must evaluate an exponent tower from the top, down. Therefore $5^{4^{3^{2}}} = 5^{4^{9}} = 5^{262144}$ and $2^{3^{4^{5}}} = 2^{3^{1024}}$ .

Akeel Howell - 4 years, 4 months ago

Large Exponents

3 solutions

0 pending reports