Answer :
Let's analyze each of the given pairs of expressions to determine if they are equivalent.
### Choice A: \(12^{2/7}\) and \((\sqrt{12})^7\)
To check for equivalency, we can rewrite the second expression using the properties of exponents.
- \(\sqrt{12} = 12^{1/2}\), so \((\sqrt{12})^7 = (12^{1/2})^7 = 12^{7 \times 1/2} = 12^{7/2}\).
Now, compare \(12^{2/7}\) and \(12^{7/2}\).
Clearly, \(12^{2/7}\) is not equal to \(12^{7/2}\).
Thus, the expressions in choice A are not equivalent.
### Choice B: \((\sqrt[3]{125})^9\) and \(125^{9/3}\)
To check for equivalency, we can rewrite the second expression using the properties of exponents.
- \(\sqrt[3]{125} = 125^{1/3}\), so \((\sqrt[3]{125})^9 = (125^{1/3})^9 = 125^{9 \times 1/3} = 125^{9/3}\).
Now, compare \((125^{1/3})^9\) and \(125^{9/3}\).
They both simplify to \(125^{9/3}\).
Thus, the expressions in choice B are equivalent.
### Choice C: \(4^{1/5}\) and \((\sqrt{4})^5\)
To check for equivalency, we can rewrite the second expression using the properties of exponents.
- \(\sqrt{4} = 4^{1/2}\), so \((\sqrt{4})^5 = (4^{1/2})^5 = 4^{5 \times 1/2} = 4^{5/2}\).
Now, compare \(4^{1/5}\) and \(4^{5/2}\).
Clearly, \(4^{1/5}\) is not equal to \(4^{5/2}\).
Thus, the expressions in choice C are not equivalent.
### Choice D: \(8^{9/2}\) and \((\sqrt{8})^9\)
To check for equivalency, we can rewrite the second expression using the properties of exponents.
- \(\sqrt{8} = 8^{1/2}\), so \((\sqrt{8})^9 = (8^{1/2})^9 = 8^{9 \times 1/2} = 8^{9/2}\).
Now, compare \(8^{9/2}\) and \(8^{9/2}\).
They are clearly equal.
Thus, the expressions in choice D are equivalent.
### Conclusion
The pairs of equivalent expressions are:
- Choice B: \((\sqrt[3]{125})^9\) and \(125^{9/3}\)
- Choice D: [tex]\(8^{9/2}\)[/tex] and [tex]\((\sqrt{8})^9\)[/tex]
### Choice A: \(12^{2/7}\) and \((\sqrt{12})^7\)
To check for equivalency, we can rewrite the second expression using the properties of exponents.
- \(\sqrt{12} = 12^{1/2}\), so \((\sqrt{12})^7 = (12^{1/2})^7 = 12^{7 \times 1/2} = 12^{7/2}\).
Now, compare \(12^{2/7}\) and \(12^{7/2}\).
Clearly, \(12^{2/7}\) is not equal to \(12^{7/2}\).
Thus, the expressions in choice A are not equivalent.
### Choice B: \((\sqrt[3]{125})^9\) and \(125^{9/3}\)
To check for equivalency, we can rewrite the second expression using the properties of exponents.
- \(\sqrt[3]{125} = 125^{1/3}\), so \((\sqrt[3]{125})^9 = (125^{1/3})^9 = 125^{9 \times 1/3} = 125^{9/3}\).
Now, compare \((125^{1/3})^9\) and \(125^{9/3}\).
They both simplify to \(125^{9/3}\).
Thus, the expressions in choice B are equivalent.
### Choice C: \(4^{1/5}\) and \((\sqrt{4})^5\)
To check for equivalency, we can rewrite the second expression using the properties of exponents.
- \(\sqrt{4} = 4^{1/2}\), so \((\sqrt{4})^5 = (4^{1/2})^5 = 4^{5 \times 1/2} = 4^{5/2}\).
Now, compare \(4^{1/5}\) and \(4^{5/2}\).
Clearly, \(4^{1/5}\) is not equal to \(4^{5/2}\).
Thus, the expressions in choice C are not equivalent.
### Choice D: \(8^{9/2}\) and \((\sqrt{8})^9\)
To check for equivalency, we can rewrite the second expression using the properties of exponents.
- \(\sqrt{8} = 8^{1/2}\), so \((\sqrt{8})^9 = (8^{1/2})^9 = 8^{9 \times 1/2} = 8^{9/2}\).
Now, compare \(8^{9/2}\) and \(8^{9/2}\).
They are clearly equal.
Thus, the expressions in choice D are equivalent.
### Conclusion
The pairs of equivalent expressions are:
- Choice B: \((\sqrt[3]{125})^9\) and \(125^{9/3}\)
- Choice D: [tex]\(8^{9/2}\)[/tex] and [tex]\((\sqrt{8})^9\)[/tex]
