Is the product of #3\sqrt 2 and 8\sqrt 18# rational or irrational?

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Answer 1 · 2017-11-13T16:05:35

See a solution process below:

First, rewrite the expression as:
#3sqrt(2) xx 8sqrt(18) =>#

#(3 xx 8)sqrt(2)sqrt(18) =>#

#24sqrt(2)sqrt(18)#

Next, we can use this rule for radicals to combine the radicals in the expression:

#sqrt(color(red)(a)) * sqrt(color(blue)(b)) = sqrt(color(red)(a) * color(blue)(b))#

#24sqrt(color(red)(2)) * sqrt(color(blue)(18)) =>#

#24sqrt(color(red)(2) * color(blue)(18)) =>#

#24sqrt(36) =>#

#24 * 6 =>#

#144#

A rational number can be written as a fraction of 2 integers:

#144 = 144/1#

Therefore the expression in the problem is rational.