Question

The value of sqrt(7 + 2sqrt(10)) is​

Answer 1

Answer:

There is a standard method to solve nested radicals:

We wish that 7+210−−√−−−−−−−−√ is a perfect square - in the form a2+2ab+b2 . After this, we can group the integer and non-integer parts together, so a2+b2=7 , and 2ab=210−−√ .

We can find a,b by looking at the factors of 10 . Since 10 is a small number, there aren’t too many to just guess. If a=1–√ , b =10−−√ , then a2+b2=11 . However, when a=2–√ , b=5–√ , then a2+b2=7 , which is what we wanted. Therefore, 7+210−−√−−−−−−−−√=(2–√+5–√)2−−−−−−−−−−√=(2–√+5–√).