To simplify the expression \( (256\sqrt{10})^{551213} \), we can start by breaking it down using the properties of exponents.
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Rewrite \( 256 \) as a power of \( 2 \): \[ 256 = 2^8 \]
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Rewrite \( \sqrt{10} \) as a power: \[ \sqrt{10} = 10^{1/2} \]
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Substitute these values into the original expression: \[ (256\sqrt{10})^{551213} = (2^8 \cdot 10^{1/2})^{551213} \]
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Use the property of exponents \( (a \cdot b)^n = a^n \cdot b^n \): \[ = (2^8)^{551213} \cdot (10^{1/2})^{551213} \]
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Apply the power of a power property \( (a^m)^n = a^{mn} \): \[ = 2^{8 \cdot 551213} \cdot 10^{(1/2) \cdot 551213} \]
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Perform the multiplication: \[ 8 \cdot 551213 = 4409704 \] \[ (1/2) \cdot 551213 = 275606.5 \]
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Substitute back into the expression: \[ = 2^{4409704} \cdot 10^{275606.5} \]
Thus, the simplified expression is: \[ 2^{4409704} \cdot 10^{275606.5} \]
This is the equivalent expression in its simplified form.
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