Let's simplify each expression step by step before ordering them.
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Expression 1: \((-2)^{-8} \cdot (2)^{11}\)
\[ (-2)^{-8} = \frac{1}{(-2)^8} = \frac{1}{256} \]
Since \((-2)^8 = (2^8) = 256\), we have: \[ (-2)^{-8} \cdot (2)^{11} = \frac{1}{256} \cdot 2^{11} = \frac{2^{11}}{256} \] Now calculating \(2^{11} = 2048\): \[ (-2)^{-8} \cdot (2)^{11} = \frac{2048}{256} = 8 \]
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Expression 2: \((\frac{1}{2})^3\)
\[ (\frac{1}{2})^3 = \frac{1}{2^3} = \frac{1}{8} \]
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Expression 3: \(2^6 - \frac{2^4}{2^2}\)
\[ 2^6 = 64 \quad \text{and} \quad \frac{2^4}{2^2} = 2^{4-2} = 2^2 = 4 \] Thus, \[ 2^6 - \frac{2^4}{2^2} = 64 - 4 = 60 \]
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Expression 4: \(\frac{1}{4^{-2}}\)
\[ 4^{-2} = \frac{1}{4^2} = \frac{1}{16} \quad \text{so} \quad \frac{1}{4^{-2}} = \frac{1}{\frac{1}{16}} = 16 \]
Now we have the simplified values:
- Expression 1: \(8\)
- Expression 2: \(\frac{1}{8}\)
- Expression 3: \(60\)
- Expression 4: \(16\)
Now let's order these values from least to greatest:
\[ \frac{1}{8} < 8 < 16 < 60 \]
Putting it all together, the order from least to greatest is: \[ (1/2)^3 < (-2)^{-8} \cdot (2)^{11} < 1/4^{-2} < 2^6 - \frac{2^4}{2^2} \]
Thus, the final order of the expressions is:
- \((\frac{1}{2})^3\)
- \((-2)^{-8} \cdot (2)^{11}\)
- \(\frac{1}{4^{-2}}\)
- \(2^6 - \frac{2^4}{2^2}\)