Let's simplify each of the given expressions step-by-step.
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\( 4^{3/2} \): \[ 4^{3/2} = (4^{1/2})^3 = (2)^3 = 8 \]
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\( 243^{3/5} \): \[ 243 = 3^5 \Rightarrow 243^{3/5} = (3^5)^{3/5} = 3^{(5 \cdot \frac{3}{5})} = 3^3 = 27 \]
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\( 25^{1/2} \): \[ 25^{1/2} = \sqrt{25} = 5 \]
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\( 27^{4/3} \cdot 27^{-1} \): \[ 27^{4/3} \cdot 27^{-1} = 27^{(4/3 - 1)} = 27^{(4/3 - 3/3)} = 27^{1/3} = \sqrt[3]{27} = 3 \]
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\( (\sqrt{2})^3 \cdot 2^{-1/2} \): \[ (\sqrt{2})^3 = (2^{1/2})^3 = 2^{(1/2 \cdot 3)} = 2^{3/2} \] Now combining with \( 2^{-1/2} \): \[ 2^{3/2} \cdot 2^{-1/2} = 2^{(3/2 - 1/2)} = 2^{2/2} = 2^1 = 2 \]
Now let's summarize the simplified results for each expression:
- \( 4^{3/2} = 8 \)
- \( 243^{3/5} = 27 \)
- \( 25^{1/2} = 5 \)
- \( 27^{4/3} \cdot 27^{-1} = 3 \)
- \( (\sqrt{2})^3 \cdot 2^{-1/2} = 2 \)
So, the final simplified results are:
- \( 8 \)
- \( 27 \)
- \( 5 \)
- \( 3 \)
- \( 2 \)