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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]
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