- Translate the problem requirements: We need to find the value of the square of the sum of two nested radical expressions: \((\sqrt{15 - 4\sqrt{14}} + \sqrt{15 + 4\sqrt{14}})^2\). This means we'll expand this expression and simplify the nested radicals.
- Expand the squared expression using basic algebra: Apply the formula \((a + b)^2 = a^2 + 2ab + b^2\) to break down the complex expression into manageable parts.
- Simplify the nested radicals by recognizing perfect square patterns: Look for ways to express the expressions under the square roots as perfect squares, which will allow us to eliminate the nested radicals.
- Calculate the final result: Combine all simplified terms to get the numerical answer.
Execution of Strategic Approach
1. Translate the problem requirements
Let's break down what we're being asked to find. We have an expression that looks complicated: \((\sqrt{15 - 4\sqrt{14}} + \sqrt{15 + 4\sqrt{14}})^2\)
In plain English, we're taking two square root expressions, adding them together, and then squaring the whole thing. The tricky part is that each square root contains another square root inside it - these are called "nested radicals."
Our goal is to find the numerical value of this expression, and we need to match it to one of the answer choices: 28, 30, 32, 34, or 36.
Process Skill: TRANSLATE - We're converting the complex mathematical notation into a clear understanding of what we need to accomplish.
2. Expand the squared expression using basic algebra
Now let's use a familiar pattern to expand this expression. When we square a sum like \((a + b)^2\), we get \(a^2 + 2ab + b^2\).
Let's call \(\sqrt{15 - 4\sqrt{14}} = a\) and \(\sqrt{15 + 4\sqrt{14}} = b\)
So our expression becomes: \(a^2 + 2ab + b^2\)
Let's calculate each piece:
- \(a^2 = (\sqrt{15 - 4\sqrt{14}})^2 = 15 - 4\sqrt{14}\)
- \(b^2 = (\sqrt{15 + 4\sqrt{14}})^2 = 15 + 4\sqrt{14}\)
- \(2ab = 2 \times \sqrt{15 - 4\sqrt{14}} \times \sqrt{15 + 4\sqrt{14}}\)
For the middle term 2ab, we can use the fact that \(\sqrt{x} \times \sqrt{y} = \sqrt{xy}\):
\(2ab = 2\sqrt{(15 - 4\sqrt{14})(15 + 4\sqrt{14})}\)
This looks like the difference of squares pattern: \((x - y)(x + y) = x^2 - y^2\)
So: \((15 - 4\sqrt{14})(15 + 4\sqrt{14}) = 15^2 - (4\sqrt{14})^2 = 225 - 16(14) = 225 - 224 = 1\)
Therefore: \(2ab = 2\sqrt{1} = 2\)
3. Simplify the nested radicals by recognizing perfect square patterns
Now let's put our pieces together:
\(a^2 + 2ab + b^2 = (15 - 4\sqrt{14}) + 2 + (15 + 4\sqrt{14})\)
Notice something beautiful happening here: the \(-4\sqrt{14}\) and \(+4\sqrt{14}\) terms cancel each other out!
So we get: \(15 - 4\sqrt{14} + 2 + 15 + 4\sqrt{14} = 15 + 2 + 15 = 32\)
Process Skill: SIMPLIFY - We recognized that the nested radical terms would cancel when we expanded the expression, making the calculation much simpler than it initially appeared.
4. Calculate the final result
Our final answer is 32.
Let's verify this makes sense: we started with a complex expression involving nested radicals, but through careful algebraic expansion, all the complicated radical terms canceled out, leaving us with simple integers that sum to 32.
Looking at our answer choices: A. 28, B. 30, C. 32, D. 34, E. 36
Our answer of 32 corresponds to choice C.
Final Answer
The value of \((\sqrt{15 - 4\sqrt{14}} + \sqrt{15 + 4\sqrt{14}})^2 = 32\)
The correct answer is C.
Common Faltering Points
Errors while devising the approach
- Attempting to simplify nested radicals individually: Students often try to simplify \(\sqrt{15 - 4\sqrt{14}}\) and \(\sqrt{15 + 4\sqrt{14}}\) separately before squaring, which is extremely difficult and time-consuming. They may waste valuable time trying to recognize these as perfect squares or attempting complex algebraic manipulations that lead nowhere.
- Missing the strategic insight to expand first: Students may not recognize that expanding \((a + b)^2 = a^2 + 2ab + b^2\) will naturally eliminate the nested radicals through cancellation. Instead, they might attempt more complicated approaches like rationalizing denominators or other advanced radical techniques.
- Fear of the nested radical complexity: The intimidating appearance of nested radicals may cause students to assume the problem requires advanced techniques, leading them to overcomplicate their approach rather than recognizing that basic algebraic expansion will work.
Errors while executing the approach
- Incorrect application of the difference of squares: When calculating \((15 - 4\sqrt{14})(15 + 4\sqrt{14})\), students may incorrectly apply the difference of squares formula. Common errors include forgetting to square the coefficient 4, calculating \((4\sqrt{14})^2\) as \(4\sqrt{14}\) instead of \(16(14) = 224\), or making arithmetic mistakes in \(15^2 - 224\).
- Sign errors during expansion: Students may make sign errors when expanding \((a + b)^2\) or when combining terms like \((15 - 4\sqrt{14}) + 2 + (15 + 4\sqrt{14})\). They might incorrectly handle the negative and positive radical terms, failing to recognize that \(-4\sqrt{14}\) and \(+4\sqrt{14}\) cancel each other out.
- Arithmetic mistakes in final calculation: Even after correctly setting up the problem, students may make basic arithmetic errors when adding \(15 + 2 + 15 = 32\), or they might incorrectly calculate intermediate steps like \(225 - 224 = 1\).
Errors while selecting the answer
No likely faltering points.
