# How do you simplify the cubed root of 150?

##color(green)(“Depending on the interpretation of the question:”)##

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##color(blue)(750sqrt(6)” “) color(brown)(” As an exact value (decimals are not always exact)”)##

Braking the question down into its component parts:

Cube##color(red)(“d”) -> (?)^3##
root of 150##->(sqrt(150))^3##

If the question had read: “cube root of 150 ” we would have ##root(3)(150)##

Although, it is not common for people just say “root” for “square root”. However, I have come across it being used in that way.

Thus there could be a contradiction in meaning of the question!
‘~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
##color(blue)(“Solution for ‘cubed root'”)##

Consider the prime factor tree of 150

We observe that 150 can be broken down to ##2xx3xx5^2##

So ##sqrt(150)=sqrt(2xx3xx5^2) = 5sqrt(6)## giving

##(sqrt(150))^3 = (5sqrt(6))^3##

This gives us

##5sqrt(6)xx5sqrt(6)xx5sqrt(6)##

##5^3xx(sqrt(6))^2xxsqrt(6)##

##125xx6xxsqrt(6)##

##color(blue)(750sqrt(6)” “)color(brown)(” As an exact value (decimals are not always exact”)##
‘~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

##color(blue)(“Solution for ‘cube root'”)##

##root(3)(150)##

George is correct. This can not be simplified any further.

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