To determine which expression is equivalent to [tex]\(\left(\frac{1}{16}\right)^{-4}\)[/tex], let's analyze each option step by step.
First, we compute [tex]\(\left(\frac{1}{16}\right)^{-4}\)[/tex]:
Recall the exponent rule for negative exponents: [tex]\(a^{-b} = \frac{1}{a^b}\)[/tex].
Applying this rule:
[tex]\[
\left(\frac{1}{16}\right)^{-4} = \left(\frac{16}{1}\right)^4 = 16^4
\][/tex]
Now we must check which option matches [tex]\(16^4\)[/tex]. Let's evaluate each option:
1. Option 1: [tex]\(-(16^4)\)[/tex]
- Calculating [tex]\( -(16^4) \)[/tex]:
[tex]\[
-(16^4) = -65536
\][/tex]
2. Option 2: [tex]\(16^4\)[/tex]
- Calculating [tex]\( 16^4 \)[/tex]:
[tex]\[
16^4 = 65536
\][/tex]
3. Option 3: [tex]\(\sqrt[4]{\frac{1}{16}}\)[/tex]
- Calculating [tex]\(\sqrt[4]{\frac{1}{16}}\)[/tex]:
[tex]\[
\sqrt[4]{\frac{1}{16}} = \left(\frac{1}{16}\right)^{\frac{1}{4}} = \frac{1}{16^{\frac{1}{4}}} = \frac{1}{2}
\][/tex]
4. Option 4: [tex]\(-\left(\frac{1}{16}\right)^{-4}\)[/tex]
- Calculating [tex]\(-\left(\frac{1}{16}\right)^{-4}\)[/tex]:
[tex]\[
-\left(\frac{1}{16}\right)^{-4} = -16^4 = -65536
\][/tex]
Comparing these results with [tex]\(\left(\frac{1}{16}\right)^{-4}\)[/tex]:
- [tex]\(\left(\frac{1}{16}\right)^{-4} = 65536\)[/tex]
- Option 1: [tex]\(-(16^4) = -65536\)[/tex] (not equivalent)
- Option 2: [tex]\(16^4 = 65536\)[/tex] (equivalent)
- Option 3: [tex]\(\sqrt[4]{\frac{1}{16}} = 0.5\)[/tex] (not equivalent)
- Option 4: [tex]\(-\left(\frac{1}{16}\right)^{-4} = -65536\)[/tex] (not equivalent)
Therefore, the expression that is equivalent to [tex]\(\left(\frac{1}{16}\right)^{-4}\)[/tex] is [tex]\(16^4\)[/tex].
So, the correct answer is:
[tex]\[
\boxed{16^4}
\][/tex]
