What is the the value of sqrt(6+sqrt(20))?

The answer has to be expressed in the form of surds.

1 Answer
Dec 29, 2017

sqrt(6+sqrt(20))=1+sqrt(5)

Explanation:

Here is one way to solve it.

Assume that sqrt(6+sqrt(20))=a+sqrt(b) where a and b are nonnegative integers.

Then, squaring both sides, 6+sqrt(20)=a^2+2asqrt(b)+b. Equating coefficients by the rationality of the terms, we find
{(a^2+b=6),(2asqrt(b)=sqrt(20)=2sqrt(5)):}

From the second equation, we have a^2b=5. Multiply both sides of the first equation by b to get a^2b+b^2=6b, or b^2-6b+5=(b-5)(b-1)=0.

The solutions of this quadratic equation are b=1 or 5, but, when b=1, a=sqrt(5).

Thus, the only solution for integers a and b is a=1,b=5.

So, we have sqrt(6+sqrt(20))=1+sqrt(5).