Answer: \(x=2^8\) or \(256\)
Solution:
Give: \(x^x=2^{2048}\)
Step 1: Write \(\mathbf{2048}\) as product of its prime
| 2 | 2048 |
| 2 | 1024 |
| 2 | 512 |
| 2 | 256 |
| 2 | 128 |
| 2 | 64 |
| 2 | 32 |
| 2 | 16 |
| 2 | 8 |
| 2 | 4 |
| 2 | 2 |
| 1 |
Therefore, \(2048=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2=2^{11}\)
Step 2: Use the laws of exponent and write the expression in RHS in form of \(\mathbf{a^a}\).
\(x^x=2^{2048}\)
\(\Rightarrow x^x=2^{2^{11}}\)
\(\Rightarrow x^x=2^{2^{3+8}}\)
\(\Rightarrow x^x=2^{2^{3}\cdot 2^{8}}\)     [\(∵ a^{m+n}=a^m\times a^n\)]
\(\Rightarrow x^x=2^{8\cdot 2^{8}}\)
\(\Rightarrow x^x=(2^8){2^{8}}\)       [\(∵ a^{mn}=(a^m)^n\)]
\(\Rightarrow x=2^8\)
\(\Rightarrow x=256\)
Hence, the value of \(\mathbf{x}\) which satisfy the equation \(\mathbf{x^x=2^{2048}}\) is \(\mathbf{2^8}\) or \(\mathbf{256}\).