Step-by-step explanation:
To find the value of \( \sqrt{\frac{\sqrt{2}-1}{\sqrt{2}+1}} \), let's first simplify the expression:
\( \sqrt{\frac{\sqrt{2}-1}{\sqrt{2}+1}} \)
We can rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator:
\( \sqrt{\frac{(\sqrt{2}-1)(\sqrt{2}-1)}{(\sqrt{2}+1)(\sqrt{2}-1)}} \)
This simplifies to:
\( \sqrt{\frac{2 - 2\sqrt{2} + 1}{2 - 1}} \)
\( \sqrt{\frac{3 - 2\sqrt{2}}{1}} \)
Now, we can further simplify this by splitting the square root:
\( \sqrt{3 - 2\sqrt{2}} \)
This can be expressed as:
\( \sqrt{(\sqrt{2})^2 - 2(\sqrt{2}) + 1^2} \)
\( \sqrt{(\sqrt{2} - 1)^2} \)
\( \sqrt{2} - 1 \)
So, the value of \( \sqrt{\frac{\sqrt{2}-1}{\sqrt{2}+1}} \) is \( \sqrt{2} - 1 \).
