Is it possible to multiply and simplify #\frac { 3} { 5x } \root [ 3] { 375a ^ { 8} b }#? If so, how?

1 Answer
smendyka
Jul 31, 2017

See a solution process below. The version of the expression you select depends on how they person asking the question wants the expression simplified.

Explanation:

We can rewrite this expression as:

#3/(5x)root(3)(125a^6 * 3a^2b)#

We can now use this rule for radicals to simplify the radical:

#root(n)(color(red)(a) * color(blue)(b)) = root(n)(color(red)(a)) * root(n)(color(blue)(b))#

#3/(5x)root(3)(color(red)(125a^6) * color(blue)(3a^2b)) =>#

#3/(5x)root(3)(color(red)(125a^6))root(3)(color(blue)(3a^2b)) =>#

#(3/(5x) * 5a^2)root(3)(color(blue)(3a^2b)) =>#

#(3/(color(red)(cancel(color(black)(5)))x) * color(red)(cancel(color(black)(5)))a^2)root(3)(color(blue)(3a^2b)) =>#

#(3a^2)/xroot(3)(color(blue)(3a^2b))#

If necessary, we can go further:

#(3a^2root(3)(3a^2 * b))/x =>#

#(3a^2root(3)(3a^2)root(3)(b))/x =>#

#(3a^2(3a^2)^(1/3)root(3)(b))/x =>#

#((3a^2)^1(3a^2)^(1/3)root(3)(b))/x =>#

#((3a^2)^(1 + 1/3)root(3)(b))/x =>#

#((3a^2)^(3/3 + 1/3)root(3)(b))/x =>#

#((3a^2)^(4/3)root(3)(b))/x =>#

#((3a^2)^(4 * 1/3)root(3)(b))/x =>#

#(((3a^2)^4)^(1/3)root(3)(b))/x =>#

#((81a^8)^(1/3)root(3)(b))/x =>#

#(root(3)(81a^8)root(3)(b))/x =>#

#root(3)(81a^8 * b)/x =>#

#root(3)(81a^8b)/x =>#

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