Question

Evaluate the following:

`(sqrt(3^4) xx root(3)(27))/(root(4)(81)) + (1/2)^-2`

Answer 1

Given: `(sqrt(3^4) xx root(3)(27))/(root(4)(81)) + (1/2)^-2`

Step-wise calculation:

1. Simplify each part:

`sqrt(3^4) = (3^4)^(1//2)`

`sqrt(3^4) = 3^(4 xx 1/2)`

`sqrt(3^4) = 3^2`

`sqrt(3^4) = 9`

`root(3)(27) = 27^(1//3)`   ...(Since 27 = 3³)

`27^(1//3) = (3^3)^(1//3)`

`27^(1//3) = 3^(3 xx 1/3)`

`27^(1//3) = 3^1`

`27^(1//3) = 3`

`root(4)(81) = 81^(1//4)`   ...(Since 81 = 3⁴)

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

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

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

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

2. Substitute these simplified results back into the expression:

`(9 xx 3)/3 + (1/2)^-2`

3. Simplify the fraction:

`27/3 = 9`

4. Simplify the power term:

`(1/2)^-2 = (2)^2`

`(1/2)^-2 = 4`

5. Add the two results:

9 + 4 = 13