Question

Evaluate the following:

`(root(4)(2^8) xx root(5)(243) xx 5^0)/root(3)(216)`

Answer 1

Given: `(root(4)(2^8) xx root(5)(243) xx 5^0)/root(3)(216)`

Step-wise calculation:

1. Compute each root and power:

`root(4)(2^8) = (2^8)^(1//4)`

`root(4)(2^8) = 2^(8 xx 1/4)`

`root(4)(2^8) = 2^2`

`root(4)(2^8) = 4`

243 = 3⁵

⇒ `root(5)(243) = (3^5)^(1//5)`

⇒ `root(5)(243) = 3^(5 xx 1/5)`

⇒ `root(5)(243) = 3^1`

⇒ `root(5)(243) = 3`

5⁰ = 1

216 = 6³

⇒ `root(3)(216) = (6^3)^(1//3)`

⇒ `root(3)(216) = 6^(3 xx 1/3)`

⇒ `root(3)(216) = 6^1`

⇒ `root(3)(216) = 6`

2. Substitute these values into the expression:

`(4 xx 3 xx 1)/6 = 12/6`

`(4 xx 3 xx 1)/6 = 2`