How do you simplify #sqrt6(sqrt3+6)#?

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Answer 1 · 2015-10-18T11:21:37

#3sqrt(2) * (1 + 2sqrt(3))#

Yourfirst step is to expand the paranthesis by using the distributive property of multiplication.

That is, you can distribute #sqrt(6)# to to both the terms that are currently in the paranthesis

#color(red)(sqrt(6)) * (sqrt(3) + 6) = color(red)(sqrt(6)) * sqrt(3) + color(red)(sqrt(6)) * 6#

Now, use the product property of radicals to write

#sqrt(6) * sqrt(3) = sqrt(6 * 3) = sqrt(18)#

The trick now is to realize that #18# can be written as a product between a perfect square and another number

#18 = 9 * 2 = 3^2 * 2#

This means that the expression can be written as

#sqrt(18) + 6sqrt(6) = sqrt(3^2 * 2) + 6sqrt(6)#

#= sqrt(3^2) * sqrt(2) + 6sqrt(6)#

# = 3sqrt(2) + 6sqrt(6)#

We're not done yet. Notice that you can write

#sqrt(6) = sqrt(2 * 3) = sqrt(2) * sqrt(3)#

This means that the expression becomes

#3sqrt(2) + 6 * sqrt(2) * sqrt(3)#

Use #3sqrt(2)# as a common factor to get

#3sqrt(2) + 6 * sqrt(2) * sqrt(3) = color(green)(3sqrt(2) * (1 + 2sqrt(3)))#