Evaluate the following?

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5 Answers
Oct 27, 2016

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

Explanation:

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

Oct 27, 2016

7/11

Explanation:

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

Oct 27, 2016

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

Explanation:

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 6th 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

Oct 27, 2016

7/11

Explanation:

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

Oct 27, 2016

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