If \(x^x=2^{2048}\), find \(x\).

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 \(\mathbf{256}\).