To determine which of the expressions is equivalent to \( 32^{1.4} \), we can simplify each option step by step.
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Option 1: \( \sqrt[7]{32^5} \)
\[ \sqrt[7]{32^5} = (32^5)^{1/7} = 32^{5/7} \]
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Option 2: \( \sqrt[5]{32^7} \)
\[ \sqrt[5]{32^7} = (32^7)^{1/5} = 32^{7/5} \]
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Option 3: \( 32^1 + 32^4 \)
\[ 32^1 + 32^4 = 32 + 32^4 \] This does not simplify to a single exponent.
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Option 4: \( 32 + 1 \times 32^4 \)
\[ 32 + 1 \times 32^4 = 32 + 32^4 \] Similar to option 3.
Now we will convert \( 1.4 \) to a fraction for easier comparison:
\[ 1.4 = \frac{14}{10} = \frac{7}{5} \]
Now, we compare the fractional exponents:
- \( 32^{1.4} = 32^{7/5} \)
- From our evaluations:
- \( \sqrt[5]{32^7} = 32^{7/5} \)
Hence, the expression that is equivalent to \( 32^{1.4} \) is: \[ \sqrt[5]{32^7} \]
So, the correct answer is: \[ \boxed{\sqrt[5]{32^7}} \]