How do you simplify #-6sqrt80 + 2sqrt75 - 8sqrt245 - 14sqrt108#?

1 Answer
Jul 15, 2017

Answer:

See a solution process below:

Explanation:

First, rewrite the radicals as:

#-6sqrt(16 * 5) + 2sqrt(25 * 3) - 8sqrt(49 * 5) - 14sqrt(36 * 3)#

Next, use this rule for radicals to simplify the radicals:

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

#-6sqrt(color(red)(16) * color(blue)(5)) + 2sqrt(color(red)(25) * color(blue)(3)) - 8sqrt(color(red)(49) * color(blue)(5)) - 14sqrt(color(red)(36) * color(blue)(3)) =>#

#-6sqrt(color(red)(16))sqrt(color(blue)(5)) + 2sqrt(color(red)(25))sqrt(color(blue)(3)) - 8sqrt(color(red)(49))sqrt(color(blue)(5)) - 14sqrt(color(red)(36))sqrt(color(blue)(3)) =>#

#(-6 * 4sqrt(color(blue)(5))) + (2 * 5sqrt(color(blue)(3))) - (8 * 7sqrt(color(blue)(5))) - (14 * 6sqrt(color(blue)(3))) =>#

#-24sqrt(color(blue)(5)) + 10sqrt(color(blue)(3)) - 56sqrt(color(blue)(5)) - 84sqrt(color(blue)(3))#

Then group and combine like terms:

#-24sqrt(color(blue)(5)) - 56sqrt(color(blue)(5)) + 10sqrt(color(blue)(3)) - 84sqrt(color(blue)(3)) =>#

#(-24 - 56)sqrt(5) + (10 - 84)sqrt(3) =>#

#-80sqrt(5) + (-74)sqrt(3) =>#

#-80sqrt(5) - 74sqrt(3)#