What is the rationalising factor of the given number?
2+sqrt(7+2sqrt(10)
2 Answers
Explanation:
I will assume that you are looking for a radical conjugate, that is an expression which when multiplied by the given expression gives a rational product.
Given:
2+sqrt(7+2sqrt(10))
First we should check whether the expression
Note that in https://socratic.org/s/aTtiPKas I found that given:
sqrt(p+qsqrt(r))" " withp, q, r > 0
then if
sqrt(p+qsqrt(r)) = sqrt(2p+2s)/2+sqrt(2p-2s)/2
In our example, putting
s = sqrt(p^2-q^2r) = sqrt(7^2-2^2(10)) = sqrt(49-40) = sqrt(9) = 3
So:
sqrt(7+2sqrt(10)) = sqrt(2(7)+2(3))/2+sqrt(2(7)-2(3))/2
color(white)(sqrt(7+2sqrt(10))) = sqrt(20)/2+sqrt(8)/2
color(white)(sqrt(7+2sqrt(10))) = sqrt(5)+sqrt(2)
So:
2+sqrt(7+2sqrt(10)) = 2+sqrt(5)+sqrt(2)
Since this involves two square roots, a suitable radical conjugate is formed by multiplying the variants of the expression
(2+sqrt(5)-sqrt(2))(2-sqrt(5)-sqrt(2))(2-sqrt(5)+sqrt(2))
=(2+sqrt(5)-sqrt(2))((2-sqrt(5))^2-(sqrt(2))^2)
=(2+sqrt(5)-sqrt(2))((4-4sqrt(5)+5)-2)
=(2+sqrt(5)-sqrt(2))(7-4sqrt(5))
=7(2+sqrt(5)-sqrt(2))-4sqrt(5)(2+sqrt(5)-sqrt(2))
=(14+7sqrt(5)-7sqrt(2))-(8sqrt(5)+20-4sqrt(10))
=-6-sqrt(5)-7sqrt(2)+4sqrt(10)
Note that this is negative, so, let's negate it to get the slightly more attractive radical conjugate:
6+sqrt(5)+7sqrt(2)-4sqrt(10)
As a check, let's multiply this by
(2+sqrt(5)+sqrt(2))(6+sqrt(5)+7sqrt(2)-4sqrt(10))
=2(6+sqrt(5)+7sqrt(2)-4sqrt(10))+sqrt(5)(6+sqrt(5)+7sqrt(2)-4sqrt(10))+sqrt(2)(6+sqrt(5)+7sqrt(2)-4sqrt(10))
=(12+2sqrt(5)+14sqrt(2)-8sqrt(10))+(6sqrt(5)+5+7sqrt(10)-20sqrt(2))+(6sqrt(2)+sqrt(10)+14-8sqrt(5))
=31
Here's another way to simplify...
Explanation:
One way of simplifying
The other zeros will be the variants:
-sqrt(7+2sqrt(10)) ," "sqrt(7-2sqrt(10)) ," " and" "-sqrt(7-2sqrt(10))
The quartic of which these are zeros can be expressed as:
(x^2-7)^2-40
= x^4-14x^2+9
= x^4-6x^2+9-8x^2
= (x^2-3)^2-(2sqrt(2)x)^2
= (x^2-2sqrt(2)x-3)(x^2+2sqrt(2)x-3)
= (x^2-2sqrt(2)x+2-5)(x^2+2sqrt(2)x+2-5)
= ((x-sqrt(2))^2-(sqrt(5))^2)((x+sqrt(2))^2-(sqrt(5))^2)
= (x-sqrt(2)-sqrt(5))(x-sqrt(2)+sqrt(5))(x+sqrt(2)-sqrt(5))(x+sqrt(2)+sqrt(5))
Hence zeros:
The greatest of these is
So:
sqrt(7+2sqrt(10)) = sqrt(2)+sqrt(5)
From this point we can proceed as my other answer to multiply:
(2+sqrt(5)-sqrt(2))(2-sqrt(5)-sqrt(2))(2-sqrt(5)+sqrt(2))