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# Quick Help

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Let a and b be positive real numbers such that a^b = b^a and b = 9a. Then a can be expressed in the form[ m sqrt (n)] where m and n are positive integers, and n is as small as possible. Find m + n

Thanks Loves

Apr 4, 2020

#1
+20876
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ab  =  ba

log(ab)  =  log(ba)

b·log(a)  =  a·log(b)

Since  b = 9a:

9a·log(a)  =  a·log(9a)

Divide both sides by a:

9·log(a)  =  log(9a)

log(a9)  =  log(9a)

a9  =  9a

Either a = 1  (which won't work)

or:         a8  =  9

a  =  91/8

a  =  (32)1/8

a  =  31/4

and  b  =  9·31/4

Which means that I can't get the answer that you want!

Apr 4, 2020
#2
+111321
+2

a^b  =  b^a

Take  the log of each side

b log a =  a log b

(b/a)  = log b  / log a

[( 9a)  /a ] =  log b  /log a

9  = log b / log a

Which implies  that

log a b  = 9

So

a^9  = b

a^9  =  9a

a^8  =  9      take  the  8th root of  each side

a =  9^(1/8) = (3^2)^(1/8)  =  3^(2/8)  =  3^(1/4)  =  4√ 3

This is what geno  found and I'm assuming that it is the  form you  want

So

m + n  =   7

Apr 4, 2020
#3
+11
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Thanks You guys Sooooo much

WITH MUCH LOVE

- C

Apr 5, 2020