Question
After Cheer Coach Moore completed grade checks and verified other eligibility requirements, there were 256 individuals remaining to try out for the 8 positions on the Sunnyville High School Varsity Cheer Squad. The individuals will go through three weeks of training, then will be face several rounds of competition. After each round, half of the individuals will be eliminated from the try-outs.
8=__(__)^x
8=__(__)^x
Answers
GPT-4o mini
To find the value of \( x \) in the equation \( 8 = 256 \cdot \left(\frac{1}{2}\right)^x \), we first simplify the equation.
Notice that:
\[
256 = 2^8 \quad \text{(since } 256 = 2^8\text{)}
\]
\[
8 = 2^3 \quad \text{(since } 8 = 2^3\text{)}
\]
So, we can rewrite the equation as:
\[
2^3 = 2^8 \cdot \left(\frac{1}{2}\right)^x
\]
Since \(\left(\frac{1}{2}\right)^x\) can be rewritten as \(2^{-x}\), we have:
\[
2^3 = 2^8 \cdot 2^{-x}
\]
This simplifies to:
\[
2^3 = 2^{8 - x}
\]
Since the bases are the same, we can set the exponents equal to each other:
\[
3 = 8 - x
\]
Now, solve for \( x \):
\[
x = 8 - 3 = 5
\]
Thus, the value of \( x \) is:
\[
\boxed{5}
\]
Notice that:
\[
256 = 2^8 \quad \text{(since } 256 = 2^8\text{)}
\]
\[
8 = 2^3 \quad \text{(since } 8 = 2^3\text{)}
\]
So, we can rewrite the equation as:
\[
2^3 = 2^8 \cdot \left(\frac{1}{2}\right)^x
\]
Since \(\left(\frac{1}{2}\right)^x\) can be rewritten as \(2^{-x}\), we have:
\[
2^3 = 2^8 \cdot 2^{-x}
\]
This simplifies to:
\[
2^3 = 2^{8 - x}
\]
Since the bases are the same, we can set the exponents equal to each other:
\[
3 = 8 - x
\]
Now, solve for \( x \):
\[
x = 8 - 3 = 5
\]
Thus, the value of \( x \) is:
\[
\boxed{5}
\]