How do you simplify the cubed root of 150?

1 Answer
Tony B
Mar 27, 2016

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

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

Explanation:

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
Tony B

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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