How do you evaluate #2 root3 { 54} - \root[ 3] { 24} - 3\root [ 3] { 16}#?

1 Answer
namralos
May 5, 2017

Simplify all of the radicals first, then combine like terms.

Explanation:

First, factor the radicands completely into prime numbers

#54 = (2)(3)(3)(3)#
#24 = (2)(2)(2)(3)#
#16 = (2)(2)(2)(2)#

Since these are cube roots, every perfect cube extracts as a single factor. So, in the case of 54, #(3)(3)(3)# extracts as 3. The other factors remain inside their respective radicals.

#root3(54) = 3root3(2)#
#root3(24) = 2root3(3)#
#root3(16) = 2root3(2)#

Our expression simplifies as

#2root3(54) - root3(24)-3root3(16) = 6root3(2) - 2root3(3) - 6root3(2)#

#= - 2root3(3)#

Observe that cube roots of 2 and cube roots of 3 would not have combined with one another. That fact is not important in this problem, though, since all of the cube roots of 2 have canceled out to zero.

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