Question

`\sqrt{4 + \sqrt{4 + \sqrt{4 + x { \\ }^(2) } } }`

\sqrt{4 + \sqrt{4 + \sqrt{4 + x { \\ }^{2} } } }
give me the full answer on notebook

Answer 1

The original expression simplifies to
`\(\sqrt{4 + √(4 + x^2)}\).`

To simplify the given expression
`\(\sqrt{4 + \sqrt{4 + √(4 + x^2)}} \)`, we can break it down step by step. Let
`\(y = √(4 + x^2)\).`

1. Substitute \(y\) into the innermost square root:

`\[ \sqrt{4 + \sqrt{4 + √(4 + x^2)}} = \sqrt{4 + √(4 + y)} \]`

2. Substitute \(y\) into the next layer:

`\[ \sqrt{4 + √(4 + y)} = √(4 + y) \]`

3. Substitute \(y\) one last time:

`\[ √(4 + y) = \sqrt{4 + √(4 + x^2)} \]`

So, the simplified expression is
`\(\sqrt{4 + √(4 + x^2)}\).`

Now, substitute y back in to get the final expression in terms of x:

`\[ \sqrt{4 + √(4 + x^2)} = \sqrt{4 + \sqrt{4 + √(4 + x^2)}} \]`