Let's analyze the expression [tex]\(\left(\frac{1}{16}\right)^{-4}\)[/tex] step by step.
1. Understanding the Negative Exponent:
An exponent of [tex]\(-4\)[/tex] means taking the reciprocal of the base and then raising it to the power of 4. Mathematically, for any base [tex]\(a\)[/tex], [tex]\(a^{-n} = \frac{1}{a^n}\)[/tex].
Applying this rule to our expression:
[tex]\[
\left(\frac{1}{16}\right)^{-4} = \left(\frac{16}{1}\right)^4 = 16^4
\][/tex]
2. Simplify the Equivalent Expression:
Now we need to compute [tex]\(16^4\)[/tex]. We can break this down as:
[tex]\[
16^4 = (16 \times 16 \times 16 \times 16)
\][/tex]
However, instead of manually calculating this, we recognize that it equals [tex]\(65536\)[/tex].
3. Compare with Given Options:
- [tex]\(-(16)^4\)[/tex]: This is the negative of [tex]\(16^4\)[/tex] and therefore not correct.
- [tex]\(16^4\)[/tex]: This matches exactly with our derived expression and its result.
- [tex]\(\sqrt[4]{\frac{1}{16}}\)[/tex]: This is the fourth root of [tex]\(\frac{1}{16}\)[/tex], which is not equivalent to [tex]\(\left(\frac{1}{16}\right)^{-4}\)[/tex].
- [tex]\(-\left(\frac{1}{16}\right)^{-4}\)[/tex]: This introduces an unnecessary negative sign and is not correct.
Therefore, the correct expression equivalent to [tex]\(\left(\frac{1}{16}\right)^{-4}\)[/tex] is:
[tex]\[
\boxed{16^4}
\][/tex]
