To simplify the expression [tex]\(\left(4 x^4\right)^{\frac{1}{2}}\)[/tex], we can break it down into simpler steps:
1. Consider the expression inside the parenthesis:
[tex]\[ 4 x^4 \][/tex]
2. Apply the exponent of [tex]\(\frac{1}{2}\)[/tex] to the entire expression:
[tex]\[ (4 x^4)^{\frac{1}{2}} \][/tex]
This can be rewritten as:
[tex]\[ (4^{\frac{1}{2}}) \cdot (x^4)^{\frac{1}{2}} \][/tex]
3. Simplify each part separately:
- For the term [tex]\(4^{\frac{1}{2}}\)[/tex]:
[tex]\[ 4^{\frac{1}{2}} \][/tex]
This represents the square root of 4:
[tex]\[ \sqrt{4} = 2 \][/tex]
- For the term [tex]\((x^4)^{\frac{1}{2}}\)[/tex]:
[tex]\[ (x^4)^{\frac{1}{2}} \][/tex]
Using the property of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[ x^{4 \cdot \frac{1}{2}} = x^2 \][/tex]
4. Multiply the simplified parts together:
[tex]\[
(4^{\frac{1}{2}}) \cdot (x^4)^{\frac{1}{2}} = 2 \cdot x^2
\][/tex]
Thus, the simplified expression is:
[tex]\[ 2 x^2 \][/tex]
Hence, the correct answer from the given options is:
[tex]\[ \boxed{2 x^2} \][/tex]
