How do you simplify ( sqrt 2 + sqrt 5)^4?
1 Answer
Apr 11, 2016
Explanation:
Note that
So let us square
(sqrt(2)+sqrt(5))^2
=(sqrt(2))^2+2(sqrt(2))(sqrt(5))+(sqrt(5))^2
=2+2sqrt(10)+5
=7+2sqrt(10)
Then:
(7+2sqrt(10))^2
=7^2+2(7)(2sqrt(10))+(2sqrt(10))^2
=49+28sqrt(10)+40
=89+28sqrt(10)
Check
Let's check the calculation a different way.
From the Binomial Theorem:
(a+b)^4 = a^4+4a^3b+6a^2b^2+4ab^3+b^4
So:
(sqrt(2)+sqrt(5))^4
=(sqrt(2))^4+4(sqrt(2))^3(sqrt(5))+6(sqrt(2))^2(sqrt(5))^2+4(sqrt(2))(sqrt(5))^3+(sqrt(5))^4
=4+8sqrt(10)+60+20sqrt(10)+25
=89+28sqrt(10)
