Question

How do you evaluate `\root[ 3] { 16} + 4\root [ 3] { 27} + \sqrt { 81}`?

Answer 1

`21+2root(3)2`

Explanation:

1) The first term `root(3)16` is the only problem because 16 is not a perfect cube.

So factor it to get a cube root out of it.

2) Then factor all the other terms as cubes or squares

`color(white)(....)root(3)16color(white)(..)+color(white)(.)4root(3)27  color(white)(.)+color(white)(.)sqrt81`

`root(3)((2)^3*2)  +  4root(3)(3^3)  +  sqrt((9)^2)`

3) Find the cube roots or the square root and write them outside
`2root(3)(2) + 4*3 + 9`

4) Evaluate the expression
`4*3 + 9 +2root(3)(2)`
`color(white)(.)12  + 9 + 2root(3)(2)`
`color(white)(......)21    + 2root(3)(2)`

Answer:
`21 + 2root(3)(2)`