To determine which expression is equivalent to [tex]\(80^{\frac{1}{4}} x\)[/tex], let's analyze each given option one by one and compare them to [tex]\(80^{\frac{1}{4}} x\)[/tex].
### Understanding the Original Expression
First, we need to interpret the original expression correctly:
[tex]\[ 80^{\frac{1}{4}} x \][/tex]
This means [tex]\( (80^{\frac{1}{4}}) \cdot x \)[/tex]. To be clear, this is the fourth root of [tex]\(80\)[/tex] times [tex]\(x\)[/tex], not [tex]\(80\)[/tex] raised to the power of [tex]\(\frac{1}{4} x\)[/tex]. The given options should be compared considering this understanding.
### Option 1: [tex]\(\left(\frac{80}{4}\right)^x\)[/tex]
Let's simplify [tex]\(\left(\frac{80}{4}\right)^x\)[/tex]:
[tex]\[ \left(\frac{80}{4}\right)^x = (20)^x \][/tex]
This is not equivalent to [tex]\(80^{\frac{1}{4}} x\)[/tex].
### Option 2: [tex]\(\sqrt[4]{80}^x\)[/tex]
This can be rewritten using exponents:
[tex]\[ \sqrt[4]{80}^x = (80^{\frac{1}{4}})^x = 80^{\frac{x}{4}} \][/tex]
Again, this is not equivalent to [tex]\(80^{\frac{1}{4}} x\)[/tex].
### Option 3: [tex]\(\sqrt[x]{80^4}\)[/tex]
Rewriting this with exponents:
[tex]\[ \sqrt[x]{80^4} = (80^4)^{\frac{1}{x}} = 80^{\frac{4}{x}} \][/tex]
This is not equivalent to [tex]\(80^{\frac{1}{4}} x\)[/tex].
### Option 4: [tex]\(\left(\frac{80}{x}\right)^4\)[/tex]
Simplifying this expression:
[tex]\[ \left(\frac{80}{x}\right)^4 = \frac{80^4}{x^4} \][/tex]
This is also not equivalent to [tex]\(80^{\frac{1}{4}} x\)[/tex].
### Correct Choice Identification
None of the provided options are directly equivalent to [tex]\(80^{\frac{1}{4}} x\)[/tex].
The closest similar form is:
[tex]\[ \sqrt[4]{80} \cdot x = 80^{\frac{1}{4}} \cdot x \][/tex]
Which can be written as:
[tex]\[ \boxed{\sqrt[4]{80} \cdot x} \][/tex]
Though not exactly present among the given choices, possibly due to a formatting issue or typographical error in the problem statement, this would be the correct equivalent expression.
The correct interpretation among provided options is not distinctively responding directly to [tex]\(80^{\frac{1}{4}} x\)[/tex] from algebraic fundamentals.
To summarize, the correct equivalence with [tex]\(80^{\frac{1}{4}} x\)[/tex] stands as:
[tex]\[ \boxed{\sqrt[4]{80} \cdot x} \][/tex]
However, it puts the best understanding for all associativity unless further settings might refine any choice interpretations.
