To simplify the expression \( (256 \sqrt{10})^{551213} \), we can use the properties of exponents and radicals.
First, let's express 256 as a power of 2:
\[ 256 = 2^8 \]
Therefore, we have:
\[ 256 \sqrt{10} = 2^8 \cdot \sqrt{10} = 2^8 \cdot (10^{1/2}) \]
We can express \( \sqrt{10} \) as \( 10^{1/2} \), and since \( 10 = 2 \cdot 5 \), we can rewrite it as:
\[ \sqrt{10} = (2 \cdot 5)^{1/2} = 2^{1/2} \cdot 5^{1/2} \]
So now we can combine everything:
\[ 256 \sqrt{10} = 2^8 \cdot 2^{1/2} \cdot 5^{1/2} = 2^{8 + 1/2} \cdot 5^{1/2} = 2^{8.5} \cdot 5^{0.5} = 2^{17/2} \cdot 5^{1/2} \]
Now our expression can be rewritten as:
\[ (256 \sqrt{10})^{551213} = (2^{17/2} \cdot 5^{1/2})^{551213} \]
Now apply the power of a product property:
\[ (2^{17/2} \cdot 5^{1/2})^{551213} = (2^{17/2})^{551213} \cdot (5^{1/2})^{551213} \]
Next, apply the exponent rule \( (a^m)^n = a^{m \cdot n} \):
\[ (2^{17/2})^{551213} = 2^{(17/2) \cdot 551213} \] \[ (5^{1/2})^{551213} = 5^{(1/2) \cdot 551213} \]
Therefore, we get:
\[ (256 \sqrt{10})^{551213} = 2^{(17/2) \cdot 551213} \cdot 5^{(1/2) \cdot 551213} \]
So, the final equivalent expression is:
\[ 2^{\frac{17 \cdot 551213}{2}} \cdot 5^{\frac{551213}{2}} \]
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