Question

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

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

Answer 1

`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)`.