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

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

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