How do you simplify #(sqrt [6] + 2sqrt [2])(4sqrt[6] - 3sqrt2)#?

Question

Savannah Anderson · Accepted Answer

#12+5sqrt12#

Given: #(sqrt6+2sqrt2)(4sqrt6-3sqrt2)#.

Use the #"FOIL"# theorem, which states that #(a+b)(c+d)=ac+ad+bc+bd#.

Thus, we obtain:

#=sqrt6*4sqrt6-3sqrt2*sqrt6+2sqrt2*4sqrt6-2sqrt2*3sqrt2#

#=4*6-3sqrt12+8sqrt12-6*2#

#=24-12+5sqrt12#

#=12+5sqrt12#

Eli Assouline · Answer

#color(crimson)(=> 2 (6 - 5 sqrt 3)# #(sqrt 6 + 2 sqrt 2) (4 sqrt 6 - 3 sqrt 2)# #=> sqrt 6 * 4 sqrt 6 + 2 sqrt 2 * 4 sqrt 6 - sqrt 6 * 3 sqrt 2 - 2 sqrt 2 * 3 sqrt 2# #=> 4 * 6 + 8 sqrt 12 - 3 sqrt 12 - 6 * 2# #=> 24 - 12 + 8 sqrt 12 - 3 sqrt 12# #=> 12 + 5 sqrt (4 * 3)# #=> 12 - 10 sqrt 3# #color(crimson)(=> 2 (6 - 5 sqrt 3)#

Ethan Adkins · Answer

undefined To simplify the expression (sqrt[6] + 2sqrt[2])(4sqrt[6] - 3sqrt[2]), you can use the distributive property. Multiply each term in the first parentheses by each term in the second parentheses.

(sqrt[6] + 2sqrt[2])(4sqrt[6] - 3sqrt[2]) = 4sqrt[6] * sqrt[6] + 4sqrt[6] * (-3sqrt[2]) + 2sqrt[2] * sqrt[6] + 2sqrt[2] * (-3sqrt[2])

Simplifying further, we have:

4sqrt[6] * sqrt[6] = 4 * 6 = 24
4sqrt[6] * (-3sqrt[2]) = -12sqrt[12]
2sqrt[2] * sqrt[6] = 2sqrt[12]
2sqrt[2] * (-3sqrt[2]) = -6sqrt[4]

Combining like terms, we get:

24 - 12sqrt[12] + 2sqrt[12] - 6sqrt[4]

Simplifying the square roots:

24 - 12sqrt[4 * 3] + 2sqrt[4 * 3] - 6sqrt[2 * 2]

24 - 12 * 2sqrt[3] + 2 * 2sqrt[3] - 6 * 2

24 - 24sqrt[3] + 4sqrt[3] - 12

Combining like terms again:

-12 - 20sqrt[3]

Therefore, the simplified expression is -12 - 20sqrt[3].