Certainly! Let's evaluate each of the given expressions step-by-step to determine which one is equivalent to [tex]\(\left(\frac{1}{16}\right)\)[/tex].
### Option 1: [tex]\(-(16)^4\)[/tex]
First, we calculate [tex]\(16^4\)[/tex]:
[tex]\[ 16^4 = 65536 \][/tex]
Next, we apply the negative sign:
[tex]\[ -(65536) = -65536 \][/tex]
So, [tex]\(-(16)^4 = -65536\)[/tex].
### Option 2: [tex]\(16^4\)[/tex]
We already calculated this in the previous step:
[tex]\[ 16^4 = 65536 \][/tex]
### Option 3: [tex]\(\sqrt[4]{\frac{1}{16}}\)[/tex]
This can be rewritten using exponent notation:
[tex]\[ \sqrt[4]{\frac{1}{16}} = \left(\frac{1}{16}\right)^{\frac{1}{4}} \][/tex]
Evaluating this:
[tex]\[ \left(\frac{1}{16}\right)^{\frac{1}{4}} = \left(16^{-1}\right)^{\frac{1}{4}} = 16^{-\frac{1}{4}} \][/tex]
[tex]\[ 16^{-\frac{1}{4}} = \left(2^4\right)^{-\frac{1}{4}} = 2^{-1} = \frac{1}{2} \][/tex]
So, [tex]\(\sqrt[4]{\frac{1}{16}} = 0.5\)[/tex].
### Option 4: [tex]\(-\left(\frac{1}{16}\right)^{-4}\)[/tex]
First, we calculate [tex]\(\left(\frac{1}{16}\right)^{-4}\)[/tex]:
[tex]\[ \left(\frac{1}{16}\right)^{-4} = \left(16^{-1}\right)^{-4} = 16^{4} = 65536 \][/tex]
Next, we apply the negative sign:
[tex]\[ -65536 \][/tex]
So, [tex]\(-\left(\frac{1}{16}\right)^{-4} = -65536\)[/tex].
### Comparing Results
The value we seek is [tex]\(\frac{1}{16}\)[/tex].
[tex]\[ \frac{1}{16} = 0.0625 \][/tex]
Comparing all computed values:
- Option 1: [tex]\(-65536\)[/tex]
- Option 2: [tex]\(65536\)[/tex]
- Option 3: [tex]\(0.5\)[/tex]
- Option 4: [tex]\(-65536\)[/tex]
None of these values match [tex]\(\left(\frac{1}{16}\right) = 0.0625\)[/tex].
Therefore, none of the given expressions are equivalent to [tex]\(\left(\frac{1}{16}\right)\)[/tex].
