Answer :
To determine which expressions are equivalent to [tex]\(\sqrt[3]{128}^x\)[/tex], we'll need to analyze and simplify each given expression carefully.
1. Given expression to compare: [tex]\(\left(\sqrt[3]{128}\right)^x\)[/tex]
We know that [tex]\( \sqrt[3]{128} \)[/tex] can be written as [tex]\( 128^{\frac{1}{3}} \)[/tex]. Thus, [tex]\(\left(\sqrt[3]{128}\right)^x\)[/tex] is equivalent to:
[tex]\[ \left(128^{\frac{1}{3}}\right)^x = 128^{\frac{x}{3}} \][/tex]
2. Checking the expressions:
- [tex]\( 128^{\frac{x}{3}} \)[/tex]:
This matches exactly with [tex]\( 128^{\frac{x}{3}} \)[/tex]. Therefore, this expression is equivalent.
- [tex]\( 128^{\frac{3}{x}} \)[/tex]:
This represents a different form and does not match [tex]\( 128^{\frac{x}{3}} \)[/tex]. Hence, this is not equivalent.
- [tex]\( (4 \sqrt[3]{2})^x \)[/tex]:
First, rewrite [tex]\( 4 \sqrt[3]{2} \)[/tex]:
[tex]\[ 4 \sqrt[3]{2} = 4 \times 2^{\frac{1}{3}} = 2^2 \times 2^{\frac{1}{3}} = 2^{2 + \frac{1}{3}} = 2^{\frac{6}{3} + \frac{1}{3}} = 2^{\frac{7}{3}} \][/tex]
Therefore,
[tex]\[ (4 \sqrt[3]{2})^x = \left(2^{\frac{7}{3}}\right)^x = 2^{\frac{7x}{3}} \][/tex]
This does not match [tex]\( 128^{\frac{x}{3}} \)[/tex].
- [tex]\( \left(4\left(2^{\frac{1}{3}}\right)\right)^x \)[/tex]:
Let's rewrite this expression:
[tex]\[ 4 \left(2^{\frac{1}{3}}\right) = 4 \times 2^{\frac{1}{3}} = 2^2 \times 2^{\frac{1}{3}} = 2^{\frac{6}{3} + \frac{1}{3}} = 2^{\frac{7}{3}} \][/tex]
So:
[tex]\[ \left(4 \left(2^{\frac{1}{3}}\right)\right)^x = \left(2^{\frac{7}{3}}\right)^x = 2^{\frac{7x}{3}} \][/tex]
Once again, this does not match [tex]\( 128^{\frac{x}{3}} \)[/tex].
- [tex]\( (2 \sqrt[3]{4})^x \)[/tex]:
Let's rewrite [tex]\( 2 \sqrt[3]{4} \)[/tex]:
[tex]\[ 2 \sqrt[3]{4} = 2 \times \sqrt[3]{4} = 2 \times 4^{\frac{1}{3}} = 2 \times (2^2)^{\frac{1}{3}} = 2 \times 2^{\frac{2}{3}} = 2^{1 + \frac{2}{3}} = 2^{\frac{3}{3} + \frac{2}{3}} = 2^{\frac{5}{3}} \][/tex]
Thus,
[tex]\[ (2 \sqrt[3]{4})^x = \left(2^{\frac{5}{3}}\right)^x = 2^{\frac{5x}{3}} \][/tex]
This does not match [tex]\( 128^{\frac{x}{3}} \)[/tex].
Conclusion:
Only the first expression, [tex]\( 128^{\frac{x}{3}} \)[/tex], is equivalent to [tex]\( \sqrt[3]{128}^x \)[/tex]. Therefore, the correct answer is:
[tex]\[ \boxed{1} \][/tex]
1. Given expression to compare: [tex]\(\left(\sqrt[3]{128}\right)^x\)[/tex]
We know that [tex]\( \sqrt[3]{128} \)[/tex] can be written as [tex]\( 128^{\frac{1}{3}} \)[/tex]. Thus, [tex]\(\left(\sqrt[3]{128}\right)^x\)[/tex] is equivalent to:
[tex]\[ \left(128^{\frac{1}{3}}\right)^x = 128^{\frac{x}{3}} \][/tex]
2. Checking the expressions:
- [tex]\( 128^{\frac{x}{3}} \)[/tex]:
This matches exactly with [tex]\( 128^{\frac{x}{3}} \)[/tex]. Therefore, this expression is equivalent.
- [tex]\( 128^{\frac{3}{x}} \)[/tex]:
This represents a different form and does not match [tex]\( 128^{\frac{x}{3}} \)[/tex]. Hence, this is not equivalent.
- [tex]\( (4 \sqrt[3]{2})^x \)[/tex]:
First, rewrite [tex]\( 4 \sqrt[3]{2} \)[/tex]:
[tex]\[ 4 \sqrt[3]{2} = 4 \times 2^{\frac{1}{3}} = 2^2 \times 2^{\frac{1}{3}} = 2^{2 + \frac{1}{3}} = 2^{\frac{6}{3} + \frac{1}{3}} = 2^{\frac{7}{3}} \][/tex]
Therefore,
[tex]\[ (4 \sqrt[3]{2})^x = \left(2^{\frac{7}{3}}\right)^x = 2^{\frac{7x}{3}} \][/tex]
This does not match [tex]\( 128^{\frac{x}{3}} \)[/tex].
- [tex]\( \left(4\left(2^{\frac{1}{3}}\right)\right)^x \)[/tex]:
Let's rewrite this expression:
[tex]\[ 4 \left(2^{\frac{1}{3}}\right) = 4 \times 2^{\frac{1}{3}} = 2^2 \times 2^{\frac{1}{3}} = 2^{\frac{6}{3} + \frac{1}{3}} = 2^{\frac{7}{3}} \][/tex]
So:
[tex]\[ \left(4 \left(2^{\frac{1}{3}}\right)\right)^x = \left(2^{\frac{7}{3}}\right)^x = 2^{\frac{7x}{3}} \][/tex]
Once again, this does not match [tex]\( 128^{\frac{x}{3}} \)[/tex].
- [tex]\( (2 \sqrt[3]{4})^x \)[/tex]:
Let's rewrite [tex]\( 2 \sqrt[3]{4} \)[/tex]:
[tex]\[ 2 \sqrt[3]{4} = 2 \times \sqrt[3]{4} = 2 \times 4^{\frac{1}{3}} = 2 \times (2^2)^{\frac{1}{3}} = 2 \times 2^{\frac{2}{3}} = 2^{1 + \frac{2}{3}} = 2^{\frac{3}{3} + \frac{2}{3}} = 2^{\frac{5}{3}} \][/tex]
Thus,
[tex]\[ (2 \sqrt[3]{4})^x = \left(2^{\frac{5}{3}}\right)^x = 2^{\frac{5x}{3}} \][/tex]
This does not match [tex]\( 128^{\frac{x}{3}} \)[/tex].
Conclusion:
Only the first expression, [tex]\( 128^{\frac{x}{3}} \)[/tex], is equivalent to [tex]\( \sqrt[3]{128}^x \)[/tex]. Therefore, the correct answer is:
[tex]\[ \boxed{1} \][/tex]
