Question

Evaluate the following:

`(81/256)^(1/4) xx (32/243)^((-2)/5)`

Answer 1

Given expression: `(81/256)^(1/4) xx (32/243)^((-2)/5)`

Step-wise calculation:

1. Write the bases as powers of primes:

81 = 3⁴

256 = 4⁴ = (2⁸)

32 = 2⁵

243 = 3⁵ 

2. Simplify each power:

`(81/256)^(1/4) = (3^4/2^8)^(1/4)`

`(81/256)^(1/4) = (3^(4 xx 1/4))/(2^(8 xx 1/4))`

`(81/256)^(1/4) = 3^1/2^2`

`(81/256)^(1/4) = 3/4`

`(32/243)^(-2/5) = (2^5/3^5)^(-2/5)`

`(32/243)^(-2/5) = (2^(5 xx -2/5))/(3^(5 xx -2/5))`

`(32/243)^(-2/5) = 2^-2/3^-2`

`(32/243)^(-2/5) = 3^2/2^2`

`(32/243)^(-2/5) = 9/4`

3. Multiply the two simplified terms:

`3/4 xx 9/4 = 27/16`

So, the value of `(81/256)^(1/4) xx (32/243)^((-2)/5)` is `27/16`.