To simplify the expression \((256\sqrt{10})^{551213}\), we can first deal with the components inside the parentheses.
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Rewrite 256: \(256\) can be expressed as a power of \(2\). Specifically, \(256 = 2^8\).
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Rewrite \(\sqrt{10}\): The square root of \(10\) can be rewritten using exponents: \[ \sqrt{10} = 10^{1/2}. \]
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Combine inside the parentheses: Now we can rewrite the expression inside the parentheses: \[ 256\sqrt{10} = 2^8 \cdot 10^{1/2}. \]
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Raise to the power of \(551213\): Now we can distribute the exponent \(551213\): \[ (256\sqrt{10})^{551213} = (2^8 \cdot 10^{1/2})^{551213}. \] Using the property \((ab)^n = a^n b^n\), we get: \[ = (2^8)^{551213} \cdot (10^{1/2})^{551213}. \]
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Apply the power of a power property: Now apply the power of a power property \((a^m)^n = a^{m \cdot n}\): \[ = 2^{8 \cdot 551213} \cdot 10^{(1/2) \cdot 551213}. \] Simplifying the exponents: \[ = 2^{4409704} \cdot 10^{275606.5}. \]
Thus, an equivalent expression for \((256\sqrt{10})^{551213}\) is:
\[ 2^{4409704} \cdot 10^{275606.5}. \]