For which of the following values of x is sqrt(1-sqrt(2-sqrt(x))) NOT defined as a real number?

1. Translate the problem requirements: We need to find which value of x makes the nested expression \(\sqrt{1-\sqrt{2-\sqrt{x}}}\) NOT defined as a real number. A square root is not defined as a real number when the expression under the square root is negative.
2. Identify the constraint structure: Work backwards from the outermost square root to understand what conditions must be met for the expression to be defined.
3. Establish the innermost constraint: For \(\sqrt{x}\) to be real, we need \(x \geq 0\).
4. Establish the middle constraint: For \(\sqrt{2-\sqrt{x}}\) to be real, we need \(2-\sqrt{x} \geq 0\), which means \(\sqrt{x} \leq 2\).
5. Establish the outermost constraint: For \(\sqrt{1-\sqrt{2-\sqrt{x}}}\) to be real, we need \(1-\sqrt{2-\sqrt{x}} \geq 0\), which means \(\sqrt{2-\sqrt{x}} \leq 1\).
6. Test each answer choice: Substitute each value and check if any constraint is violated.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're looking for in plain English. We have a nested square root expression that looks like layers of an onion: \(\sqrt{1-\sqrt{2-\sqrt{x}}}\). We need to find which value of x makes this expression NOT work as a real number.

Think of it this way: when does a square root "break"? It breaks when we try to take the square root of a negative number. For example, \(\sqrt{-4}\) is not a real number.

So our job is to find which value of x from the answer choices creates a negative number under one of these square root signs.

Process Skill: TRANSLATE - Converting the problem language into a clear mathematical understanding

2. Identify the constraint structure

Since we have three nested square roots, we need to work backwards from the outside to understand what can go wrong. Think of it like checking a chain - each link needs to work for the whole chain to work.

Let's identify our three "links":

• Outermost: \(\sqrt{1-\sqrt{2-\sqrt{x}}}\) - this needs \(1-\sqrt{2-\sqrt{x}} \geq 0\)
• Middle: \(\sqrt{2-\sqrt{x}}\) - this needs \(2-\sqrt{x} \geq 0\)
• Innermost: \(\sqrt{x}\) - this needs \(x \geq 0\)

If any of these conditions fails, our expression is not defined as a real number.

3. Establish the innermost constraint

Starting from the inside and working out, for \(\sqrt{x}\) to be a real number, we need:

x must be greater than or equal to 0

This makes sense - we can't take the square root of a negative number and get a real result. So \(x \geq 0\).

4. Establish the middle constraint

For \(\sqrt{2-\sqrt{x}}\) to be real, we need what's under this square root to be non-negative:

2 - \(\sqrt{x}\) must be greater than or equal to 0

Rearranging this in plain English: \(\sqrt{x}\) must be less than or equal to 2
Or mathematically: \(\sqrt{x} \leq 2\)

Since we're dealing with positive values, we can square both sides: \(x \leq 4\)

5. Establish the outermost constraint

For the outermost square root \(\sqrt{1-\sqrt{2-\sqrt{x}}}\) to be real, we need:

1 - \(\sqrt{2-\sqrt{x}}\) must be greater than or equal to 0

Rearranging: \(\sqrt{2-\sqrt{x}}\) must be less than or equal to 1
Or mathematically: \(\sqrt{2-\sqrt{x}} \leq 1\)

Squaring both sides: \(2-\sqrt{x} \leq 1\)
Rearranging: \(\sqrt{x} \geq 1\)
Squaring again: \(x \geq 1\)

Process Skill: APPLY CONSTRAINTS - Systematically checking each nested condition

6. Test each answer choice

Now let's combine our constraints. For the expression to be defined as a real number, we need:

• \(x \geq 0\) (innermost constraint)
• \(x \leq 4\) (middle constraint)
• \(x \geq 1\) (outermost constraint)

Combining these: \(1 \leq x \leq 4\)

Let's test each choice:

1. x = 1: \(1 \leq 1 \leq 4\) ✓ (Expression IS defined)
2. x = 2: \(1 \leq 2 \leq 4\) ✓ (Expression IS defined)
3. x = 3: \(1 \leq 3 \leq 4\) ✓ (Expression IS defined)
4. x = 4: \(1 \leq 4 \leq 4\) ✓ (Expression IS defined)
5. x = 5: \(1 \leq 5 \leq 4\) ✗ (Expression is NOT defined)

Let's verify x = 5 step by step:

• \(\sqrt{5} \approx 2.24\)
• \(2 - \sqrt{5} \approx 2 - 2.24 = -0.24\)
• \(\sqrt{2-\sqrt{5}} = \sqrt{-0.24}\) which is NOT a real number

Process Skill: CONSIDER ALL CASES - Testing each answer choice systematically

4. Final Answer

The expression \(\sqrt{1-\sqrt{2-\sqrt{x}}}\) is NOT defined as a real number when x = 5, because this creates a negative value under the middle square root.

The answer is E.

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misunderstanding the question requirement
Students often miss that the question asks for values where the expression is NOT defined, and instead look for values where it IS defined. This leads them to find the valid range \(1 \leq x \leq 4\) but then incorrectly select an answer from within this range rather than outside it.

Faltering Point 2: Incorrect constraint setup order
Students may try to work from the outermost square root inward instead of starting from the innermost. This approach becomes confusing because the outer constraints depend on the inner expressions being valid first, leading to incomplete or incorrect constraint identification.

Faltering Point 3: Overlooking the nested structure
Students might treat this as a simple single square root problem and only consider one constraint (like \(x \geq 0\)) rather than recognizing that all three nested levels must simultaneously satisfy their respective constraints.

Errors while executing the approach

Faltering Point 1: Sign errors when rearranging inequalities
When solving \(\sqrt{2-\sqrt{x}} \leq 1\), students often make errors in the algebraic manipulation, particularly when rearranging \(2-\sqrt{x} \leq 1\) to get \(\sqrt{x} \geq 1\). The subtraction and inequality direction can be confusing.

Faltering Point 2: Incorrect squaring of inequalities
Students may forget that when squaring both sides of an inequality involving square roots, they need to ensure both sides are non-negative. For example, when going from \(\sqrt{x} \geq 1\) to \(x \geq 1\), they might not verify that both \(\sqrt{x}\) and 1 are non-negative.

Faltering Point 3: Computational errors in constraint combination
After finding individual constraints (\(x \geq 0\), \(x \leq 4\), \(x \geq 1\)), students may incorrectly combine them, perhaps getting \(0 \leq x \leq 1\) instead of \(1 \leq x \leq 4\), leading to wrong conclusions about which values work.

Errors while selecting the answer

Faltering Point 1: Selecting a value that makes the expression defined
After correctly finding that the expression is defined for \(1 \leq x \leq 4\), students might accidentally select answer choice A, B, C, or D (values that DO work) instead of E (the value that does NOT work), forgetting the 'NOT' in the question.

Faltering Point 2: Insufficient verification of the final answer
Students may not double-check their answer by substituting x = 5 back into the original expression to confirm that \(\sqrt{2-\sqrt{5}}\) involves taking the square root of a negative number, missing the opportunity to catch potential errors in their constraint analysis.