Let's go through your questions one by one:
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Which factors of the base would help simplify the expression \(16^{1/4}\)? Given the options:
- 16 = 4⋅2⋅2
- 16 = 4⋅4 (This one is helpful because \(4 = 2^2\))
- 16 = 8⋅8
- 16 = 2⋅2⋅2⋅2 (This is also useful because \(2^4 = 16\))
Correct Response: 16 = 4⋅4 (This is the most straightforward way to simplify \(16^{1/4}\) using powers)
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Given that the cube root of 27 is 3, what is \(27^{4/3}\)? Using the property \(a^{m/n} = (a^m)^{1/n}\):
- We can calculate \(27^{4/3} = (27^{4})^{1/3} = (3^3)^4)^{1/3} = 3^4 = 81\).
Correct Response: 81
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What is the correct simplification of \(25^{3/2}\)? \(25^{3/2} = (25^{1/2})^3 = (5)^3 = 125\).
Correct Response: 125
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Given that the meaning of a rational exponent can be connected to the meaning of a root, how can you rewrite \(5^{-\sqrt{3}}\) using a rational exponent? The expression \(5^{-\sqrt{3}}\) corresponds to \(5^{-3^{1/2}}\).
Correct Response: 5 superscript Start Fraction -3 End Fraction baseline (As that indicates a negative exponent and involves a square root.)
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How can you rewrite \(8^{1/2}\) using a root? This can be expressed as \(\sqrt{8}\).
Correct Response: 8–√ (Start Root 8 End Root)
Feel free to ask if you have more questions or need further clarification!