- Translate the problem requirements: We need to find the value of the expression \((\sqrt{9+\sqrt{80}} + \sqrt{9-\sqrt{80}})^2\) by expanding the squared binomial and simplifying the nested radicals.
- Recognize the algebraic expansion pattern: Apply the formula \((\mathrm{a} + \mathrm{b})^2 = \mathrm{a}^2 + 2\mathrm{ab} + \mathrm{b}^2\) to expand the squared expression.
- Simplify the individual radical terms: Calculate the squares of each radical term and identify what \(\sqrt{80}\) simplifies to for easier computation.
- Evaluate the cross-product term: Find the value of \(2\sqrt{(9+\sqrt{80})(9-\sqrt{80})}\) using the difference of squares pattern.
- Combine all terms: Add the simplified components to reach the final numerical answer.
Execution of Strategic Approach
1. Translate the problem requirements
We need to find the value of \((\sqrt{9+\sqrt{80}} + \sqrt{9-\sqrt{80}})^2\). Let's call this expression our target.
Think of this like unwrapping a present - we have a squared expression on the outside, and inside we have two square root terms that we need to add together first. The challenge is that each square root contains another square root inside it.
Our goal is to find a numerical value that matches one of the answer choices.
Process Skill: TRANSLATE
2. Recognize the algebraic expansion pattern
When we have something like \((\mathrm{a} + \mathrm{b})^2\), we know from basic algebra that this equals \(\mathrm{a}^2 + 2\mathrm{ab} + \mathrm{b}^2\).
In our case:
- a = \(\sqrt{9+\sqrt{80}}\)
- b = \(\sqrt{9-\sqrt{80}}\)
So our expression becomes:
\((\sqrt{9+\sqrt{80}})^2 + 2 \cdot \sqrt{9+\sqrt{80}} \cdot \sqrt{9-\sqrt{80}} + (\sqrt{9-\sqrt{80}})^2\)
This simplifies to:
\((9+\sqrt{80}) + 2\sqrt{(9+\sqrt{80})(9-\sqrt{80})} + (9-\sqrt{80})\)
3. Simplify the individual radical terms
Let's first look at \(\sqrt{80}\). We can break this down:
\(\sqrt{80} = \sqrt{16 \times 5} = \sqrt{16} \times \sqrt{5} = 4\sqrt{5}\)
Now our expression becomes:
\((9+4\sqrt{5}) + 2\sqrt{(9+4\sqrt{5})(9-4\sqrt{5})} + (9-4\sqrt{5})\)
Notice that the first and third terms: \((9+4\sqrt{5}) + (9-4\sqrt{5}) = 18\)
The \(4\sqrt{5}\) terms cancel out!
4. Evaluate the cross-product term
Now we need to find \(2\sqrt{(9+4\sqrt{5})(9-4\sqrt{5})}\).
This looks like the difference of squares pattern: \((\mathrm{a}+\mathrm{b})(\mathrm{a}-\mathrm{b}) = \mathrm{a}^2 - \mathrm{b}^2\)
where a = 9 and b = \(4\sqrt{5}\)
\((9+4\sqrt{5})(9-4\sqrt{5}) = 9^2 - (4\sqrt{5})^2 = 81 - 16 \times 5 = 81 - 80 = 1\)
Therefore: \(2\sqrt{(9+4\sqrt{5})(9-4\sqrt{5})} = 2\sqrt{1} = 2 \times 1 = 2\)
Process Skill: SIMPLIFY
5. Combine all terms
Putting it all together:
Original expression = 18 + 2 = 20
Final Answer
The value of \((\sqrt{9+\sqrt{80}} + \sqrt{9-\sqrt{80}})^2 = 20\)
This matches answer choice E.
Common Faltering Points
Errors while devising the approach
- Not recognizing the algebraic expansion pattern: Students might try to evaluate \(\sqrt{9+\sqrt{80}}\) and \(\sqrt{9-\sqrt{80}}\) individually first, rather than recognizing that squaring the sum \((\mathrm{a}+\mathrm{b})^2 = \mathrm{a}^2 + 2\mathrm{ab} + \mathrm{b}^2\) will lead to a much cleaner solution path.
- Getting intimidated by nested radicals: The presence of square roots within square roots can cause students to abandon systematic algebraic approaches and instead attempt complex decimal approximations or give up entirely.
- Missing the difference of squares opportunity: Students may not anticipate that the cross-product term \(\sqrt{(9+\sqrt{80})(9-\sqrt{80})}\) will simplify beautifully using the difference of squares pattern, leading them to choose more complicated solution methods.
Errors while executing the approach
- Arithmetic errors when simplifying \(\sqrt{80}\): Students might incorrectly calculate \(\sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5}\), perhaps writing it as \(2\sqrt{5}\) or making other factorization mistakes, which would throw off all subsequent calculations.
- Sign errors in the difference of squares: When calculating \((9+4\sqrt{5})(9-4\sqrt{5}) = 9^2 - (4\sqrt{5})^2\), students might forget to square the entire term \((4\sqrt{5})\) properly, writing it as \(81 - 4\sqrt{5}\) instead of \(81 - 16 \times 5 = 81 - 80\).
- Forgetting the coefficient 2 in the cross-product term: After correctly finding that \(\sqrt{(9+\sqrt{80})(9-\sqrt{80})} = 1\), students might forget to multiply by 2, giving a final answer of 19 instead of 20.
Errors while selecting the answer
- Selecting a complex expression instead of the simplified numerical value: Even after getting the correct numerical result of 20, students might second-guess themselves and choose one of the more complex algebraic expressions (like options B or C) thinking the answer "looks too simple."
