Square Root Within Square Root: √(6-√11) + √(6+√11) can be expressed as √w

[Firas]

Square Root Within Square Root: √(6-√11) + √(6+√11) can be expressed as √w

The expression √(6-√11) + √(6+√11) can be expressed as √w.

Find w, if it is a whole number.

What i thought of doing is squaring both sides but i think that will be way too complicated

 

[HallsofIvy]

If nothing else, you can do this numerically.

Approximately,
\(\displaystyle 6- \sqrt{11}= 6- 3.3166= 2.6834\) so \(\displaystyle \sqrt{6- \sqrt{11}}= \sqrt{2.6834}= 1.6381\).

\(\displaystyle 6+ \sqrt{11}= 6+ 3.3166= 9.3166\) so \(\displaystyle \sqrt{6+ \sqrt{11}}= \sqrt{9.3166}= 3.0523\).

\(\displaystyle \sqrt{6- \sqrt{11}}+ \sqrt{6+ \sqrt{11}}= 1.6381+ 3.0523= 4.6904\).

If that is to be \(\displaystyle \sqrt{w}\), what can we try now?

 

[Dr.Peterson]

The expression √(6-√11) + √(6+√11) can be expressed as √w.

Find w, if it is a whole number.

What i thought of doing is squaring both sides but i think that will be way too complicated

Try squaring both sides of the equation, just as you said, keeping everything in radical form, and then solving the resulting radical equation for w.

If you get tangled up, you might actually try generalizing the question, squaring both sides of √(a-b) + √(a+b)= √w and solving for w in terms of a and b. If nothing else, this is a little more impressive. This is what I actually did.

Show your work if you have trouble.

By the way, Halls' work will come in handy ...

 

[Denis]

Example of DrP's : sqrt(10 - 6) + sqrt(10 + 6) = sqrt(36)

Another? Here: sqrt(97 - 72) + sqrt(97 + 72) = sqrt(37636)