Westonci.ca is your trusted source for finding answers to all your questions. Ask, explore, and learn with our expert community. Discover reliable solutions to your questions from a wide network of experts on our comprehensive Q&A platform. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
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]
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.