Algebra - "Solve for x" Challenge¶
Problem Statement¶
Solve for \(x\) and \(y\). You should have coordinate pairs (\(x\), \(y\)) for each equation:
\[
x^x = 256
\]
Step-by-Step Solution¶
To solve the equation \( x^x = 256 \), follow these steps:
Step 1: Express 256 as a power of a base¶
First, express 256 as a power of 2, because it simplifies solving equations involving exponents:
\[
256 = 2^8
\]
Thus, the equation \( x^x = 256 \) can be rewritten as:
\[
x^x = 2^8
\]
Step 2: Try simple values for \( x \)¶
Check if any integer value for \( x \) makes \( x^x = 2^8 \) true.
- Try \( x = 2 \):
\[
2^2 = 4
\]
This does not match \( 2^8 \).
- Try \( x = 4 \):
\[
4^4 = (2^2)^4 = 2^{8} = 256
\]
This matches \( 2^8 \).
Step 3: Verify the solution¶
The value \( x = 4 \) satisfies the equation:
\[
4^4 = 256
\]
Thus, \( x = 4 \) is a solution.
Solution:¶
The solution to the equation \( x^x = 256 \) is:
\[
x = {\color{red} 4}
\]