To determine which expressions are equivalent to [tex]\((\sqrt[3]{128})^x\)[/tex], let's first simplify the original expression:
1. Simplify [tex]\(\sqrt[3]{128}\)[/tex]:
[tex]\[ \sqrt[3]{128} = 128^{1/3} \][/tex]
So, the expression we are working with is:
[tex]\[ (128^{1/3})^x \][/tex]
By the properties of exponents, this simplifies to:
[tex]\[ (128^{1/3})^x = 128^{(x/3)} \][/tex]
Now, let's examine each given expression one by one to see if they simplify to [tex]\(128^{(x/3)}\)[/tex]:
### Expression 1: [tex]\( 128^{\frac{x}{3}} \)[/tex]
This is exactly the same as our simplified form [tex]\(128^{(x/3)}\)[/tex].
### Expression 2: [tex]\( 128^{\frac{3}{x}} \)[/tex]
This expression is not equivalent to [tex]\(128^{(x/3)}\)[/tex]. Exponents do not match the form we are looking for.
### Expression 3: [tex]\( (4 \sqrt[3]{2})^x \)[/tex]
Simplify this expression step by step:
First, simplify inside the parenthesis:
[tex]\[ 4 \sqrt[3]{2} = 4 \cdot 2^{1/3} \][/tex]
[tex]\[ = 2^2 \cdot 2^{1/3} \][/tex]
[tex]\[ = 2^{2 + 1/3} \][/tex]
[tex]\[ = 2^{6/3 + 1/3} \][/tex]
[tex]\[ = 2^{7/3} \][/tex]
Now, raise this to the power [tex]\(x\)[/tex]:
[tex]\[ (2^{7/3})^x \][/tex]
[tex]\[ = 2^{(7x/3)} \][/tex]
Now we need to relate this to [tex]\(128 = 2^7\)[/tex] raised to the power [tex]\(x/3\)[/tex]:
[tex]\[ 128^{(1/3)x} = (2^7)^{(x/3)} = 2^{(7x/3)} \][/tex]
So, [tex]\( (4 \sqrt[3]{2})^x \)[/tex] is equivalent to [tex]\(128^{(x/3)}\)[/tex].
### Expression 4: [tex]\( \left(4\left(2^{\frac{1}{3}}\right)\right)^x \)[/tex]
Simplify this expression step by step:
First, simplify inside the parenthesis:
[tex]\[ 4 \cdot 2^{1/3} = 2^2 \cdot 2^{1/3} \][/tex]
[tex]\[ = 2^{2 + 1/3} \][/tex]
[tex]\[ = 2^{6/3 + 1/3} \][/tex]
[tex]\[ = 2^{7/3} \][/tex]
Now, raise this to the power [tex]\(x\)[/tex]:
[tex]\[ (2^{7/3})^x \][/tex]
[tex]\[ = 2^{7x/3} \][/tex]
As shown in Expression 3, this is also equivalent to [tex]\(128^{(x/3)}\)[/tex].
### Expression 5: [tex]\( (2 \sqrt[3]{4})^x \)[/tex]
Simplify this expression step by step:
First, simplify inside the parenthesis:
[tex]\[ 2 \cdot \sqrt[3]{4} = 2 \cdot 4^{1/3} \][/tex]
[tex]\[ = 2 \cdot (2^2)^{1/3} \][/tex]
[tex]\[ = 2 \cdot 2^{2/3} \][/tex]
[tex]\[ = 2^{1 + 2/3} \][/tex]
[tex]\[ = 2^{3/3 + 2/3} \][/tex]
[tex]\[ = 2^{5/3} \][/tex]
Now, raise this to the power [tex]\(x\)[/tex]:
[tex]\[ (2^{5/3})^x \][/tex]
[tex]\[ = 2^{5x/3} \][/tex]
However, [tex]\(2^{5x/3}\)[/tex] is not equivalent to [tex]\(128^{(x/3)} = 2^{7x/3}\)[/tex].
### Conclusion
The equivalent expressions are:
1. [tex]\(128^{\frac{x}{3}}\)[/tex]
2. [tex]\((4 \sqrt[3]{2})^x\)[/tex]
3. [tex]\(\left(4\left(2^{\frac{1}{3}}\right)\right)^x\)[/tex]
The three correct answers are:
1. [tex]\(128^{\frac{x}{3}}\)[/tex]
2. [tex]\((4 \sqrt[3]{2})^x\)[/tex]
3. [tex]\(\left(4\left(2^{\frac{1}{3}}\right)\right)^x\)[/tex]
