Let's find the expression equivalent to [tex]\( \sqrt[3]{8} \cdot \frac{1}{4} \cdot x \)[/tex].
First, we analyze the term [tex]\( \sqrt[3]{8} \)[/tex]. This can be rewritten using exponents as:
[tex]\[ 8^{\frac{1}{3}} \][/tex]
Next, we need to multiply this by [tex]\(\frac{1}{4}\)[/tex] and [tex]\(x\)[/tex]. This gives us:
[tex]\[ 8^{\frac{1}{3}} \cdot \frac{1}{4} \cdot x \][/tex]
This expression simplifies to:
[tex]\[ \left( 8^{\frac{1}{3}} \cdot \frac{1}{4} \right) \cdot x \][/tex]
Now, let's consider the given options and see which one matches our simplified expression:
1. [tex]\( 8^{\frac{3}{4} x} \)[/tex]
- This represents [tex]\( 8 \)[/tex] raised to the power of [tex]\(\frac{3}{4} x\)[/tex].
2. [tex]\( \sqrt[7]{8}^x \)[/tex]
- This represents the 7th root of [tex]\( 8 \)[/tex] raised to the power of [tex]\( x \)[/tex], which can be rewritten as:
[tex]\[ (8^{\frac{1}{7}})^x = 8^{\frac{x}{7}} \][/tex]
3. [tex]\( \sqrt[12]{8} \cdot x \)[/tex]
- This represents the 12th root of [tex]\( 8 \)[/tex] multiplied by [tex]\( x \)[/tex], which can be rewritten as:
[tex]\[ 8^{\frac{1}{12}} \cdot x \][/tex]
4. [tex]\( 8^{\frac{3}{4 x}} \)[/tex]
- This represents [tex]\( 8 \)[/tex] raised to the power of [tex]\(\frac{3}{4x}\)[/tex].
From these options, the one that matches our simplified expression [tex]\( 8^{\frac{1}{3}} \cdot \frac{1}{4} \cdot x \)[/tex] is:
[tex]\[ \sqrt[12]{8} \cdot x = 8^{\frac{1}{12}} \cdot x \][/tex]
Thus, the correct choice is:
[tex]\[ 2 \][/tex]
This corresponds to the expression [tex]\( \sqrt[12]{8} \cdot x \)[/tex].
