Let's solve the problem step-by-step using properties of exponents.
Given function:
[tex]\[ f(x) = 6 \left(\frac{1}{4}\right)^{2x} \][/tex]
To rewrite this function, we need to simplify the term [tex]\(\left(\frac{1}{4}\right)^{2x}\)[/tex] using exponent properties.
### Step 1: Simplify [tex]\(\left(\frac{1}{4}\right)^{2x}\)[/tex]
Recall the exponent property: [tex]\((a^m)^n = a^{m \cdot n}\)[/tex].
Rewrite [tex]\(\left(\frac{1}{4}\right)^{2x}\)[/tex] as:
[tex]\[ \left(\frac{1}{4}\right)^{2x} = \left(\left(\frac{1}{4}\right)^2\right)^x \][/tex]
### Step 2: Calculate [tex]\(\left(\frac{1}{4}\right)^2\)[/tex]
Calculate [tex]\(\left(\frac{1}{4}\right)^2\)[/tex]:
[tex]\[ \left(\frac{1}{4}\right)^2 = \frac{1}{4^2} = \frac{1}{16} \][/tex]
So, substituting back in, we have:
[tex]\[ \left(\frac{1}{4}\right)^{2x} = \left(\frac{1}{16}\right)^x \][/tex]
### Step 3: Substitute back into the original function
Now, substitute [tex]\(\left(\frac{1}{16}\right)^x\)[/tex] into the original function:
[tex]\[ f(x) = 6 \left(\frac{1}{16}\right)^x \][/tex]
This is our simplified form of the function.
### Check the choices given:
1. [tex]\( 36 \left(\frac{1}{4}\right)^x \)[/tex]
2. [tex]\( 36 \left(\frac{1}{16}\right)^x \)[/tex]
3. [tex]\( 6 \left(\frac{1}{16}\right)^x \)[/tex]
4. [tex]\( 6 \left(\frac{2}{8}\right)^x \)[/tex]
From our simplification, the correct form of the function is:
[tex]\[ f(x) = 6 \left(\frac{1}{16}\right)^x \][/tex]
So the correct answer is:
[tex]\[ \boxed{6 \left(\frac{1}{16}\right)^x} \][/tex]
