How do you simplify #(4 sqrt7 - 8 sqrt 5) (5sqrt7 + 10sqrt 5)#?

Question

Eli Assouline · Accepted Answer

See a solution process below:

To multiply these two terms you multiply each individual term in the left parenthesis by each individual term in the right parenthesis. #(color(red)(4sqrt(7)) - color(red)(8sqrt(5)))(color(blue)(5sqrt(7)) + color(blue)(10sqrt(5)))# becomes: #(color(red)(4sqrt(7)) xx color(blue)(5sqrt(7))) + (color(red)(4sqrt(7)) xx color(blue)(10sqrt(5))) - (color(red)(8sqrt(5)) xx color(blue)(5sqrt(7))) - (color(red)(8sqrt(5)) xx color(blue)(10sqrt(5)))# #(20 xx 7) + (40(color(red)(sqrt(7)) xx color(blue)(sqrt(5)))) - (40(color(red)(sqrt(5)) xx color(blue)(sqrt(7)))) - (80 xx 5)# #140 + (40(color(red)(sqrt(7)) xx color(blue)(sqrt(5)))) - (40(color(red)(sqrt(5)) xx color(blue)(sqrt(7)))) - 400# We can now group and combine like terms: #140 - 400 + (40(color(red)(sqrt(7)) xx color(blue)(sqrt(5)))) - (40(color(red)(sqrt(5)) xx color(blue)(sqrt(7))))# #(140 - 400) + (40 - 40)(color(red)(sqrt(7)) xx color(blue)(sqrt(5)))# #-260 + 0(color(red)(sqrt(7)) xx color(blue)(sqrt(5)))# #-260 + 0# #-260#

Eli Assouline · Answer

undefined To simplify the expression (4√7 - 8√5)(5√7 + 10√5), we can use the distributive property.

First, multiply the terms in the first parentheses with the terms in the second parentheses:

(4√7)(5√7) + (4√7)(10√5) - (8√5)(5√7) - (8√5)(10√5)

This simplifies to:

20√49 + 40√35 - 40√35 - 80√25

Next, simplify the square roots:

20(7) + 40√35 - 40√35 - 80(5)

This further simplifies to:

140 + 0 - 400

Finally, combine like terms:

-260