To simplify the expression \( (256 \sqrt{10})^{551213} \), we can use the properties of exponents.
First, we can express \( \sqrt{10} \) as \( 10^{1/2} \). Then we rewrite the original expression using the properties of exponents:
\[ (256 \sqrt{10})^{551213} = (256 \cdot 10^{1/2})^{551213} \]
Using the property \( (a \cdot b)^m = a^m \cdot b^m \):
\[ = 256^{551213} \cdot (10^{1/2})^{551213} \]
Now, applying the exponent to the \( 10^{1/2} \):
\[ = 256^{551213} \cdot 10^{(1/2) \cdot 551213} \]
To simplify further, let's evaluate \( 256^{551213} \). Note that \( 256 = 2^8 \):
\[ 256^{551213} = (2^8)^{551213} = 2^{8 \cdot 551213} = 2^{4409704} \]
Now, we simplify the \( 10^{(1/2) \cdot 551213} \):
\[ 10^{(1/2) \cdot 551213} = 10^{275606.5} \]
Thus, the entire expression is:
\[ (256 \sqrt{10})^{551213} = 2^{4409704} \cdot 10^{275606.5} \]
And remember that \( 10 = 2 \cdot 5 \), so
\[ 10^{275606.5} = (2 \cdot 5)^{275606.5} = 2^{275606.5} \cdot 5^{275606.5} \]
Combining all the terms gives us:
\[ = 2^{4409704} \cdot 2^{275606.5} \cdot 5^{275606.5} = 2^{4409704 + 275606.5} \cdot 5^{275606.5} \]
Calculating the exponent for \( 2 \):
\[ 4409704 + 275606.5 = 4685310.5 \]
So, compiling the entire expression:
\[ = 2^{4685310.5} \cdot 5^{275606.5} \]
This is the simplified form of the expression \( (256 \sqrt{10})^{551213} \). If needed, we can express it back in its original format or leave it in this exponential form.