How do you simplify #root3(150)*root3(20)#?

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Answer 1 · 2016-10-21T11:56:41

#10root(3)3#

Dealing with roots is the same as dealing with exponents. Except the exponents for roots are fractions. Like this:
#root(3)x=x^(1/3)#

That means we can apply the distributive property of exponents to roots as well:
#root(3)(x*y)=(x*y)^(1/3)=x^(1/3)*y^(1/3)#

Now to our problem. If we factor both numbers we get that:
#150=5*5*3*2#
#20=5*2*2#

so our problem becomes:
#root(3)(5*5*3*2)*root(3)(5*2*2)#

Then we can combine both under one root, like so:
#root(3)(5*5*3*2*5*2*2)#

As you can see there are three 5s and three 2s in there, meaning we can take them to the outside of the root like this:
#5*2root(3)3#

or even better:
#10root(3)3#