Determine which function produces the same graph as $$ f(x)=\left(8^{\frac{2}{3} x}\right)\left(16^{\frac{1}{2} x}\right) $$ $$ f(x)=4^{x} $$ $$ f(x)=4^{2 x} $$ $$ f(x)=8^{3 x} $$ $$ f(x)=16^{2 x} $$

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To determine which function produces the same graph as $$ f(x) = \left(8^{\frac{2}{3} x}\right)\left(16^{\frac{1}{2} x}\right) $$, let's simplify $$ f(x) $$:

1. **Express in terms of base 2:**
   - $$ 8 = 2^3 $$
   - $$ 16 = 2^4 $$

2. **Rewrite $$ f(x) $$ using base 2:**
   $$
   f(x) = \left(2^3\right)^{\frac{2}{3} x} \left(2^4\right)^{\frac{1}{2} x}
   $$

3. **Simplify the exponents:**
   $$
   f(x) = 2^{3 \cdot \frac{2}{3} x} \cdot 2^{4 \cdot \frac{1}{2} x} = 2^{2x} \cdot 2^{2x} = 2^{4x}
   $$

Now, let's compare this with the given options by expressing them in terms of base 2:

- $$ f(x) = 4^x = (2^2)^x = 2^{2x} $$
- $$ f(x) = 4^{2x} = (2^2)^{2x} = 2^{4x} $$
- $$ f(x) = 8^{3x} = (2^3)^{3x} = 2^{9x} $$
- $$ f(x) = 16^{2x} = (2^4)^{2x} = 2^{8x} $$

Comparing the exponents, $$ 2^{4x} $$ matches with $$ f(x) = 4^{2x} $$.

**Answer:** $$ f(x) = 4^{2 x} $$
$$ f(x) = 4^{2x} $$ produces the same graph as the original function.

Determine which function produces the same graph as \( f(x)=\left(8^{\frac{2}{3} x}\right)\left(16^{\frac{1}{2} x}\right) \) \( f(x)=4^{x} \) \( f(x)=4^{2 x} \) \( f(x)=8^{3 x} \) \( f(x)=16^{2 x} \)

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