Question

Evaluate the following?

Answer 1

`(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`

Answer 2

`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`

Answer 3

`(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 `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`

Answer 4

`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`

Answer 5

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