Answer :
Answer:
To solve \((\sqrt[4]{2})^2\) and show that it equals \(\sqrt{2}\), let's break it down step by step.
1. First, understand what \(\sqrt[4]{2}\) represents. \(\sqrt[4]{2}\) is the fourth root of 2, which can be written as:
\[
\sqrt[4]{2} = 2^{\frac{1}{4}}
\]
2. Next, we square \(\sqrt[4]{2}\):
\[
(\sqrt[4]{2})^2 = (2^{\frac{1}{4}})^2
\]
3. When raising a power to another power, we multiply the exponents. So, we multiply \(\frac{1}{4}\) by 2:
\[
(2^{\frac{1}{4}})^2 = 2^{\frac{1}{4} \times 2} = 2^{\frac{2}{4}} = 2^{\frac{1}{2}}
\]
4. The exponent \(\frac{1}{2}\) represents the square root. So:
\[
2^{\frac{1}{2}} = \sqrt{2}
\]
Therefore:
\[
(\sqrt[4]{2})^2 = \sqrt{2}
\]
This completes the proof that \((\sqrt[4]{2})^2 = \sqrt{2}\).