Answer :

Certainly, let's go through the process step-by-step to determine whether [tex]\(\sqrt{\sqrt{2}}\)[/tex] is rational or irrational.

### Step 1: Find [tex]\(\sqrt{2}\)[/tex]

Using the long division method, we determine that:
[tex]\[ \sqrt{2} \approx 1.4142135623730951 \][/tex]

### Step 2: Find [tex]\(\sqrt{\sqrt{2}}\)[/tex]

Next, we need to find [tex]\(\sqrt{\sqrt{2}}\)[/tex]. Since we know:
[tex]\[ \sqrt{2} \approx 1.4142135623730951 \][/tex]
We then take the square root of this value:
[tex]\[ \sqrt{1.4142135623730951} \approx 1.189207115002721 \][/tex]

### Step 3: Determine Whether [tex]\(\sqrt{\sqrt{2}}\)[/tex] is Rational or Irrational

A number is rational if it can be expressed as a fraction [tex]\(\frac{p}{q}\)[/tex], where [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are integers and [tex]\(q \neq 0\)[/tex]. Conversely, if a number cannot be expressed as such a fraction, it is termed irrational.

Now, we examine:
[tex]\[ \sqrt{\sqrt{2}} \approx 1.189207115002721 \][/tex]

The decimal form here is non-terminating and non-repeating. This suggests that [tex]\(\sqrt{\sqrt{2}}\)[/tex] cannot be expressed as a fraction of integers, which is a strong indication that it is irrational.

### Conclusion

[tex]\(\sqrt{\sqrt{2}} \approx 1.189207115002721\)[/tex] is irrational, as it cannot be written as a fraction of integers.

Therefore, [tex]\(\sqrt{\sqrt{2}}\)[/tex] is irrational.