Evaluate the following?

Using the rules of exponents that #(a/b)^x = a^x/b^x#, and that #(a^x)^y = a^(xy)#, we have

#(117649/1771561)^(1/6) = 117649^(1/6)/1771561^(1/6)#

#=(7^6)^(1/6)/(11^6)^(1/6)#

#=7^(6*1/6)/11^(6*1/6)#

#=7^1/11^1#

#=7/11#

This is exactly the same concept as a previous question with an index of #1/8#

#rarr # find the prime factors.
#rarr # index of #1/6# means the same as #root6(color(white)n)#

#117649/1771561 = (7xx7xx7xx7xx7xx7)/(11xx11xx11xx11xx11xx11)#

#((7^6)/(11^6))^(1/6) = root6((7^6)/(11^6))#

=#7/11#

Check #117649# for its prime factorisation:

• Last digit is odd, so not divisible by #2#.
• Sum of digits is #1+1+7+6+4+9 = 28# not divisible by #3#.
• Last digit is not #0# or #5#, so not divisible by #5#
• #117649/7 = 16807#, so divisible by #7#
• #16807/7 = 2401#
• #2401/7 = 343#
• #343/7 = 49#
• #49/7 = 7#

So #117649 = 7^6#

So if the answer is exact, we require the denominator to be a perfect #6#th power too.

Let's do some approximating...

#1771561/117649 ~~ 177/12 ~~ 14#

#sqrt(14) ~~ 4#

#root(3)(4) ~~ 3/2#

So: #(14)^(1/6) ~~ 3/2#

#3/2*7 ~~ 11#

Try:

• #1771561/11 = 161051#
• #161051/11 = 14641#
• #14641/11 = 1331#
• #1331/11 = 121#
• #121/11 = 11#

So #1771561 = 11^6#

Hence:

#(117649/1771561)^(1/6) = (7^6/11^6)^(1/6) = 7/11#

Computing the rational number with fractional exponent #(u/v)^(1/6)=u^(1/6)/v^(1/6)# is determined by prime factorizing the numerator and denominator .

example of: #sqrtx = x^(1/2#:

#sqrt((4/9)) = (4/9)^(color(blue)(1/2))=(2^2)^color(blue)(1/2)/(3^2)^color(blue)(1/2)=2/3#

The rational number above #(117649/1771561)^(1/6)=(117649)^(1/6)/(1771561)^(1/6)#

Prime Factorization :
#color(blue)(117649=7xx7xx7xx7xx7xx7=7^6)#
#color(brown)(1771561=11xx11xx11xx11xx11xx11=11^6)#

#(117649/1771561)^(1/6)=(117649)^(1/6)/(1771561)^(1/6)=(color(blue)(7^6))^(1/6)/(color(brown)(11^6))^(1/6)#

Then we apply the power of a power with base #a# :
#(a^m)^(1/n)=a^(m/n)#

#(117649/1771561)^(1/6)=(7^(6/6)/11^(6/6))=7/11#

Calculator reveals the value as

0.6363636363..

#=(10^(-2)/(1-10^(-2))) 63= 63/99=7/11#

Reference:
https://socratic.org/questions/a-fraction-v-in-decimal-form-is-an-infinite-string-that-comprises-the-non-repeat#323791