Answer :
To determine which pairs of expressions are equivalent, let's analyze each pair step-by-step.
### Pair A: \((\sqrt[3]{125})^9\) and \(125^{9 / 3}\)
1. Evaluate \((\sqrt[3]{125})^9\):
- \(\sqrt[3]{125}\) means finding the cube root of 125.
- The cube root of 125 is 5 because \(5^3 = 125\).
- So, \((\sqrt[3]{125})^9 = (5)^9 = 5^9\).
2. Evaluate \(125^{9 / 3}\):
- \(125^{9 / 3} = 125^3\) since \(9 / 3 = 3\).
- \(125^3 = (5^3)^3 = 5^{33} = 5^9\).
Since both expressions simplify to \(5^9\), they are equivalent.
### Pair B: \(12^{2 / 7}\) and \((\sqrt{12})^7\)
1. Evaluate \(12^{2 / 7}\):
- This expression represents 12 raised to the power of \(2 / 7\).
2. Evaluate \((\sqrt{12})^7\):
- \(\sqrt{12}\) means the square root of 12.
- The square root of 12 is \(12^{1 / 2}\).
- So, \((\sqrt{12})^7 = (12^{1 / 2})^7\).
- Using the property of exponents, \((a^b)^c = a^{bc}\), we get \((12^{1 / 2})^7 = 12^{(1 / 2) 7} = 12^{7 / 2}\).
The expressions \(12^{2 / 7}\) and \(12^{7 / 2}\) are not equivalent.
### Pair C: \(4^{1 / 5}\) and \((\sqrt{4})^5\)
1. Evaluate \(4^{1 / 5}\):
- This expression represents 4 raised to the power of \(1 / 5\).
2. Evaluate \((\sqrt{4})^5\):
- \(\sqrt{4}\) means the square root of 4.
- The square root of 4 is 2 because \(2^2 = 4\).
- So, \((\sqrt{4})^5 = (2)^5 = 2^5\).
The expressions \(4^{1 / 5}\) and \(2^5\) are not equivalent.
### Pair D: \(8^{9 / 2}\) and \((\sqrt{8})^9\)
1. Evaluate \(8^{9 / 2}\):
- This expression represents 8 raised to the power of \(9 / 2\).
2. Evaluate \((\sqrt{8})^9\):
- \(\sqrt{8}\) means the square root of 8.
- The square root of 8 is \(8^{1 / 2}\).
- So, \((\sqrt{8})^9 = (8^{1 / 2})^9\).
- Using the property of exponents, \((a^b)^c = a^{bc}\), we get \((8^{1 / 2})^9 = 8^{(1 / 2) * 9} = 8^{9 / 2}\).
Since both expressions simplify to \(8^{9 / 2}\), they are equivalent.
### Conclusion
After analyzing each pair:
- Pair A: \((\sqrt[3]{125})^9\) and \(125^{9 / 3}\) are equivalent.
- Pair B: \(12^{2 / 7}\) and \((\sqrt{12})^7\) are not equivalent.
- Pair C: \(4^{1 / 5}\) and \((\sqrt{4})^5\) are not equivalent.
- Pair D: \(8^{9 / 2}\) and \((\sqrt{8})^9\) are equivalent.
Thus, the pairs of equivalent expressions are A and D.
### Pair A: \((\sqrt[3]{125})^9\) and \(125^{9 / 3}\)
1. Evaluate \((\sqrt[3]{125})^9\):
- \(\sqrt[3]{125}\) means finding the cube root of 125.
- The cube root of 125 is 5 because \(5^3 = 125\).
- So, \((\sqrt[3]{125})^9 = (5)^9 = 5^9\).
2. Evaluate \(125^{9 / 3}\):
- \(125^{9 / 3} = 125^3\) since \(9 / 3 = 3\).
- \(125^3 = (5^3)^3 = 5^{33} = 5^9\).
Since both expressions simplify to \(5^9\), they are equivalent.
### Pair B: \(12^{2 / 7}\) and \((\sqrt{12})^7\)
1. Evaluate \(12^{2 / 7}\):
- This expression represents 12 raised to the power of \(2 / 7\).
2. Evaluate \((\sqrt{12})^7\):
- \(\sqrt{12}\) means the square root of 12.
- The square root of 12 is \(12^{1 / 2}\).
- So, \((\sqrt{12})^7 = (12^{1 / 2})^7\).
- Using the property of exponents, \((a^b)^c = a^{bc}\), we get \((12^{1 / 2})^7 = 12^{(1 / 2) 7} = 12^{7 / 2}\).
The expressions \(12^{2 / 7}\) and \(12^{7 / 2}\) are not equivalent.
### Pair C: \(4^{1 / 5}\) and \((\sqrt{4})^5\)
1. Evaluate \(4^{1 / 5}\):
- This expression represents 4 raised to the power of \(1 / 5\).
2. Evaluate \((\sqrt{4})^5\):
- \(\sqrt{4}\) means the square root of 4.
- The square root of 4 is 2 because \(2^2 = 4\).
- So, \((\sqrt{4})^5 = (2)^5 = 2^5\).
The expressions \(4^{1 / 5}\) and \(2^5\) are not equivalent.
### Pair D: \(8^{9 / 2}\) and \((\sqrt{8})^9\)
1. Evaluate \(8^{9 / 2}\):
- This expression represents 8 raised to the power of \(9 / 2\).
2. Evaluate \((\sqrt{8})^9\):
- \(\sqrt{8}\) means the square root of 8.
- The square root of 8 is \(8^{1 / 2}\).
- So, \((\sqrt{8})^9 = (8^{1 / 2})^9\).
- Using the property of exponents, \((a^b)^c = a^{bc}\), we get \((8^{1 / 2})^9 = 8^{(1 / 2) * 9} = 8^{9 / 2}\).
Since both expressions simplify to \(8^{9 / 2}\), they are equivalent.
### Conclusion
After analyzing each pair:
- Pair A: \((\sqrt[3]{125})^9\) and \(125^{9 / 3}\) are equivalent.
- Pair B: \(12^{2 / 7}\) and \((\sqrt{12})^7\) are not equivalent.
- Pair C: \(4^{1 / 5}\) and \((\sqrt{4})^5\) are not equivalent.
- Pair D: \(8^{9 / 2}\) and \((\sqrt{8})^9\) are equivalent.
Thus, the pairs of equivalent expressions are A and D.
