To simplify the expression \( (256\sqrt{10})^{551213} \), we can use the properties of exponents and radicals.
First, we express the square root and the number 256 in terms of exponents:
- Rewrite \( \sqrt{10} \) as \( 10^{1/2} \).
- Rewrite \( 256 \) as \( 2^8 \) since \( 2^8 = 256 \).
Now, we can rewrite the expression:
\[ (256\sqrt{10})^{551213} = (2^8 \cdot 10^{1/2})^{551213} \]
Using the property of exponents \( (a \cdot b)^n = a^n \cdot b^n \):
\[ (2^8 \cdot 10^{1/2})^{551213} = (2^8)^{551213} \cdot (10^{1/2})^{551213} \]
We apply the property of exponents \( (a^m)^n = a^{m \cdot n} \):
\[ (2^8)^{551213} = 2^{8 \cdot 551213} = 2^{4409704} \] \[ (10^{1/2})^{551213} = 10^{(1/2) \cdot 551213} = 10^{275606.5} \]
Now we can combine both parts:
\[ (256\sqrt{10})^{551213} = 2^{4409704} \cdot 10^{275606.5} \]
This is the equivalent expression for \((256\sqrt{10})^{551213}\).
Thus, the final answer is:
\[ 2^{4409704} \cdot 10^{275606.5} \]