Westonci.ca is the Q&A platform that connects you with experts who provide accurate and detailed answers. Discover a wealth of knowledge from professionals across various disciplines on our user-friendly Q&A platform. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.

Which of these choices show a pair of equivalent expressions? Check all that apply.

A. [tex]$12^{2 / 7}$[/tex] and [tex]$(\sqrt[7]{12})^2$[/tex]

B. [tex]$(\sqrt[3]{125})^9$[/tex] and [tex]$125^{9 / 3}$[/tex]

C. [tex]$4^{1 / 5}$[/tex] and [tex]$(\sqrt[5]{4})$[/tex]

D. [tex]$8^{9 / 2}$[/tex] and [tex]$(\sqrt{8})^9$[/tex]


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]
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.