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.
