To simplify the expression \( (256\sqrt{10})^{551213} \), we can first break it down:
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Recognize that \( 256 \) can be expressed as a power of 2: \[ 256 = 2^8 \] So we can rewrite the expression as: \[ (256\sqrt{10})^{551213} = (2^8 \sqrt{10})^{551213} \]
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The square root can be rewritten as a fractional exponent: \[ \sqrt{10} = 10^{1/2} \] Therefore, we have: \[ (2^8 \sqrt{10})^{551213} = (2^8 \cdot 10^{1/2})^{551213} \]
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Now we apply the exponent to both components in the parentheses: \[ = (2^8)^{551213} \cdot (10^{1/2})^{551213} \]
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Using the property of exponents \( (a^m)^n = a^{m \cdot n} \): \[ = 2^{8 \cdot 551213} \cdot 10^{(1/2) \cdot 551213} \]
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Now we can calculate the exponents: \[ 8 \cdot 551213 = 4409704 \] and \[ \frac{1}{2} \cdot 551213 = 275606.5 \]
Thus, the equivalent expression is: \[ 2^{4409704} \cdot 10^{275606.5} \]
This final expression, \( 2^{4409704} \cdot 10^{275606.5} \), is the simplified equivalent of the original expression \( (256\sqrt{10})^{551213} \).
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