To determine which function has a simplified base equivalent to [tex]\(4 \sqrt[3]{4}\)[/tex], we need to simplify the bases of each function and compare them to [tex]\(4 \sqrt[3]{4}\)[/tex].
Given functions:
1. [tex]\( f(x) = 2 (\sqrt[3]{16})^x \)[/tex]
2. [tex]\( f(x) = 2 (\sqrt[3]{64})^x \)[/tex]
3. [tex]\( f(x) = 4 (\sqrt[3]{16})^{2 x} \)[/tex]
4. [tex]\( f(x) = 4 (\sqrt[3]{64})^{2 x} \)[/tex]
Now let's look at each base one at a time:
### Function 1: [tex]\( f(x) = 2 (\sqrt[3]{16})^x \)[/tex]
To simplify the base:
[tex]\[ 16^{1/3} = \sqrt[3]{16} \][/tex]
Thus, the simplified base is:
[tex]\[ 2 \times 16^{1/3} \][/tex]
The calculated value for this base is:
[tex]\[ 5.039684199579493 \][/tex]
### Function 2: [tex]\( f(x) = 2 (\sqrt[3]{64})^x \)[/tex]
To simplify the base:
[tex]\[ 64^{1/3} = \sqrt[3]{64} = 4 \][/tex]
Thus, the simplified base is:
[tex]\[ 2 \times 4 \][/tex]
The calculated value for this base is:
[tex]\[ 8.000000000000000 \][/tex]
### Function 3: [tex]\( f(x) = 4 (\sqrt[3]{16})^{2 x} \)[/tex]
To simplify the base:
[tex]\[ (\sqrt[3]{16})^2 = (16^{1/3})^2 \][/tex]
Thus, the simplified base is:
[tex]\[ 4 \times (16^{1/3})^2 \][/tex]
The calculated value for this base is:
[tex]\[ 25.398416831491193 \][/tex]
### Function 4: [tex]\( f(x) = 4 (\sqrt[3]{64})^{2 x} \)[/tex]
To simplify the base:
[tex]\[ (\sqrt[3]{64})^2 = (64^{1/3})^2 = 4^2 = 16 \][/tex]
Thus, the simplified base is:
[tex]\[ 4 \times 16 \][/tex]
The calculated value for this base is:
[tex]\[ 64.00000000000000 \][/tex]
Next, we need to compare these calculated values with the target base [tex]\( 4 \sqrt[3]{4} \)[/tex].
### Target Base: [tex]\( 4 \sqrt[3]{4} \)[/tex]
[tex]\[ 4 \times 4^{1/3} \][/tex]
The calculated value for the target base is:
[tex]\[ 6.3496042078727974 \][/tex]
### Comparison:
- [tex]\( f(x) = 2 (\sqrt[3]{16})^x \)[/tex] has a base [tex]\( 5.039684199579493 \)[/tex]
- [tex]\( f(x) = 2 (\sqrt[3]{64})^x \)[/tex] has a base [tex]\( 8.000000000000000 \)[/tex]
- [tex]\( f(x) = 4 (\sqrt[3]{16})^{2 x} \)[/tex] has a base [tex]\( 25.398416831491193 \)[/tex]
- [tex]\( f(x) = 4 (\sqrt[3]{64})^{2 x} \)[/tex] has a base [tex]\( 64.00000000000000 \)[/tex]
- The target base [tex]\( 4 \sqrt[3]{4} \)[/tex] is [tex]\( 6.3496042078727974 \)[/tex]
None of the function bases match the target base exactly. However, the closest value is from [tex]\( f(x) = 2 (\sqrt[3]{64})^x \)[/tex], which is [tex]\( 8.000000000000000 \)[/tex]. But still, there is no exact function with the simplified base [tex]\( 4 \sqrt[3]{4} \)[/tex].
