To find how many rounds it will take for only 8 individuals to remain from the original 256, we need to determine how many times we can eliminate half of the individuals until we reach 8.
Starting with \( n = 256 \), after each round where half are eliminated, the remaining number of individuals can be represented by the equation:
\[ n = \frac{256}{2^x} \]
Where:
- \( n \) is the final number of individuals (which we want to be 8),
- \( x \) is the number of rounds.
We set up the equation:
\[ 8 = \frac{256}{2^x} \]
To solve for \( x \), we first multiply both sides by \( 2^x \):
\[ 8 \cdot 2^x = 256 \]
Now, we can express 256 as a power of 2:
\[ 8 \cdot 2^x = 2^8 \]
Since 8 is equal to \( 2^3 \), we have:
\[ 2^3 \cdot 2^x = 2^8 \]
Combining the exponents on the left side gives:
\[ 2^{3 + x} = 2^8 \]
Since the bases are the same, we can set the exponents equal to each other:
\[ 3 + x = 8 \]
Solving for \( x \) gives:
\[ x = 8 - 3 = 5 \]
So, the completed equation is:
\[ 8 = \frac{256}{(2)^5} \]
Therefore, the answers are:
- __ becomes 256
- __ becomes 2
- __ becomes 5
So the filled-in equation reads:
\[ 8 = \frac{256}{(2)^5} \]
3 answers