Answer :
Let's analyze each pair of expressions to determine if they are equivalent step by step.
### Choice A:
[tex]\[ (\sqrt[3]{125})^9 \quad \text{and} \quad 125^{9 / 3} \][/tex]
- First expression: \((\sqrt[3]{125})^9\)
- The cube root of 125 is \(125^{1/3}\).
- Raising it to the 9th power: \((125^{1/3})^9 = 125^{(1/3) \cdot 9} = 125^3\).
- Second expression: \(125^{9 / 3}\)
- Simplify the exponent: \(125^{9 / 3} = 125^3\).
Both expressions simplify to \(125^3\), which means they are equivalent.
### Choice B:
[tex]\[ 12^{2 / 7} \quad \text{and} \quad (\sqrt{12})^7 \][/tex]
- First expression: \(12^{2 / 7}\)
- This is already simplified.
- Second expression: \((\sqrt{12})^7\)
- The square root of 12 is \(12^{1/2}\).
- Raising it to the 7th power: \((12^{1/2})^7 = 12^{(1/2) \cdot 7} = 12^{7/2}\).
Comparing the exponents, \(12^{2/7}\) and \(12^{7/2}\) are not equivalent.
### Choice C:
[tex]\[ 4^{1 / 5} \quad \text{and} \quad (\sqrt{4})^5 \][/tex]
- First expression: \(4^{1 / 5}\)
- This is already simplified.
- Second expression: \((\sqrt{4})^5\)
- The square root of 4 is \(4^{1/2}\).
- Raising it to the 5th power: \((4^{1/2})^5 = 4^{(1/2) \cdot 5} = 4^{5/2}\).
Comparing the exponents, \(4^{1/5}\) and \(4^{5/2}\) are not equivalent.
### Choice D:
[tex]\[ 8^{9 / 2} \quad \text{and} \quad (\sqrt{8})^9 \][/tex]
- First expression: \(8^{9 / 2}\)
- This is already simplified.
- Second expression: \((\sqrt{8})^9\)
- The square root of 8 is \(8^{1/2}\).
- Raising it to the 9th power: \((8^{1/2})^9 = 8^{(1/2) \cdot 9} = 8^{9/2}\).
Both expressions simplify to \(8^{9/2}\), which means they are equivalent.
### Conclusion
After analyzing all the pairs, we find that the equivalent expressions are found in:
- Choice A: \((\sqrt[3]{125})^9\) and \(125^{9 / 3}\)
- Choice D: \(8^{9 / 2}\) and \((\sqrt{8})^9\)
Therefore, the correct and equivalent choices are A and D.
### Choice A:
[tex]\[ (\sqrt[3]{125})^9 \quad \text{and} \quad 125^{9 / 3} \][/tex]
- First expression: \((\sqrt[3]{125})^9\)
- The cube root of 125 is \(125^{1/3}\).
- Raising it to the 9th power: \((125^{1/3})^9 = 125^{(1/3) \cdot 9} = 125^3\).
- Second expression: \(125^{9 / 3}\)
- Simplify the exponent: \(125^{9 / 3} = 125^3\).
Both expressions simplify to \(125^3\), which means they are equivalent.
### Choice B:
[tex]\[ 12^{2 / 7} \quad \text{and} \quad (\sqrt{12})^7 \][/tex]
- First expression: \(12^{2 / 7}\)
- This is already simplified.
- Second expression: \((\sqrt{12})^7\)
- The square root of 12 is \(12^{1/2}\).
- Raising it to the 7th power: \((12^{1/2})^7 = 12^{(1/2) \cdot 7} = 12^{7/2}\).
Comparing the exponents, \(12^{2/7}\) and \(12^{7/2}\) are not equivalent.
### Choice C:
[tex]\[ 4^{1 / 5} \quad \text{and} \quad (\sqrt{4})^5 \][/tex]
- First expression: \(4^{1 / 5}\)
- This is already simplified.
- Second expression: \((\sqrt{4})^5\)
- The square root of 4 is \(4^{1/2}\).
- Raising it to the 5th power: \((4^{1/2})^5 = 4^{(1/2) \cdot 5} = 4^{5/2}\).
Comparing the exponents, \(4^{1/5}\) and \(4^{5/2}\) are not equivalent.
### Choice D:
[tex]\[ 8^{9 / 2} \quad \text{and} \quad (\sqrt{8})^9 \][/tex]
- First expression: \(8^{9 / 2}\)
- This is already simplified.
- Second expression: \((\sqrt{8})^9\)
- The square root of 8 is \(8^{1/2}\).
- Raising it to the 9th power: \((8^{1/2})^9 = 8^{(1/2) \cdot 9} = 8^{9/2}\).
Both expressions simplify to \(8^{9/2}\), which means they are equivalent.
### Conclusion
After analyzing all the pairs, we find that the equivalent expressions are found in:
- Choice A: \((\sqrt[3]{125})^9\) and \(125^{9 / 3}\)
- Choice D: \(8^{9 / 2}\) and \((\sqrt{8})^9\)
Therefore, the correct and equivalent choices are A and D.
