AMC 10 · 2007 · #20

Grade 8 algebra
Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity
symmetric-polynomialsexponentsformula-substitution formula-substitution ↑ Prerequisites: symmetric-polynomials
📏 Medium solution 💡 2 insights
Problem
A number a satisfies a + a⁻¹ = 4, where a⁻¹ means the reciprocal 1/a. Find the value of a⁴ + a⁻⁴.

Pick an answer.

(A)
164
(B)
172
(C)
192
(D)
194
(E)
212

AMC 10 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Change Focus / Count the Complement

The direct route, solving a + a⁻¹ = 4 for a, gives the ugly irrational a = 2 ± √(3), and raising that to the fourth power by hand is painful. Tool #16 (Change Focus) redirects the goal: never find a at all, and instead work only with the symmetric quantities aⁿ + a⁻ⁿ. The engine that makes this work is squaring, because a · a⁻¹ = 1 makes the middle term of every square a clean constant 2. Tool #7 (Identify Subproblems) turns the leap from power 1 to power 4 into two easy rungs, a + a⁻¹ → a² + a⁻² → a⁴ + a⁻⁴. Tool #5 (Look for a Pattern) recognizes that both rungs are the exact same move — square, then subtract 2 — so the second step needs no new idea.

1STEP 1

Aim at the powers, not at a

Never solve for a. Squaring x + x⁻¹ gives x² + x⁻² plus a tidy +2, because x · x⁻¹ = 1 — and the exponent doubles.

(x + x⁻¹)² = x² + 2 x x⁻¹ + x⁻² = x² + x⁻² + 2
2STEP 2

Square once to reach power 2

Square a + a⁻¹ = 4: the left side is a² + a⁻² + 2 and the right side is 16, so a² + a⁻² = 14.

(a + a⁻¹)² = 4² → a² + a⁻² + 2 = 16 → a² + a⁻² = 14
3STEP 3

Square again to reach power 4

Repeat the move on a² + a⁻² = 14: now a⁴ + a⁻⁴ + 2 = 196, so a⁴ + a⁻⁴ = 194, choice (D).

(a² + a⁻²)² = 14² → a⁴ + a⁻⁴ + 2 = 196 → a⁴ + a⁻⁴ = 194 → (D)
Answer
194
Check with the actual value of a. The equation a + a⁻¹ = 4 becomes a² - 4a + 1 = 0, so a = 2 + √(3) (its reciprocal is 2 - √(3)). Then a² = 7 + 4√(3) and a⁻² = 7 - 4√(3), which sum to 14, matching Step 2. Squaring again, a⁴ = 97 + 56√(3) and a⁻⁴ = 97 - 56√(3), which sum to 194. The irrational parts cancel exactly, confirming a⁴ + a⁻⁴ = 194. The number is also comfortably inside the answer range and just below 14² = 196, as expected since a⁴ + a⁻⁴ is 14² minus 2.
💡Key takeaway

To find a high power of a number plus its reciprocal, don't solve for the number — just square the equation, because the reciprocals multiply to 1 and leave a clean +2 while the exponent doubles.

  • Aim at the powers, not at a
  • Square once to reach power 2
  • Square again to reach power 4

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