To simplify the expression \( (256\sqrt{10})^{551213} \), we first rewrite the components using properties of exponents.
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Rewriting 256 with exponents: We know that \( 256 = 2^8 \). Thus, we can rewrite our expression: \[ 256\sqrt{10} = 2^8 \cdot (10^{1/2}) = 2^8 \cdot 10^{1/2}. \]
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Using Properties of Exponents: Now let's raise this entire expression to the power of \( 551213 \): \[ (256\sqrt{10})^{551213} = (2^8 \cdot 10^{1/2})^{551213}. \] By using the property \( (a \cdot b)^n = a^n \cdot b^n \), we can separate the expression: \[ (2^8)^{551213} \cdot (10^{1/2})^{551213}. \]
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Apply the exponent to each factor: \[ (2^8)^{551213} = 2^{8 \cdot 551213} = 2^{4409704}, \] and \[ (10^{1/2})^{551213} = 10^{\frac{551213}{2}}. \]
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Combine the results: Thus, we have: \[ (256\sqrt{10})^{551213} = 2^{4409704} \cdot 10^{\frac{551213}{2}}. \]
In conclusion, the equivalent expression for \( (256\sqrt{10})^{551213} \) is: \[ 2^{4409704} \cdot 10^{\frac{551213}{2}}. \]