AMC 10 · 2005 · #17

Grade 8 arithmetic
Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps
exponents pattern-recognitionconvert-to-algebra ↑ Prerequisites: exponents
📏 Short solution 💡 2 insights
Problem
Four numbers a, b, c, d are hidden inside four equations: 4^a=5, 5^b=6, 6^c=7, and 7^d=8. Find the value of the product a·b·c·d.

Pick an answer.

(A)
1
(B)
$\frac{3}{2}$
(C)
2
(D)
$\frac{5}{2}$
(E)
3

AMC 10 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Change Focus / Count the Complement

Chasing a,b,c,d separately is hopeless because none of them is a nice number. Tool #16 (Change Focus) is the key move: stop asking for each exponent and instead ask what a single power, 4^a b c d, equals — because the rule (x^a)^b=x^ab means stacking the four exponents is the same as multiplying them. Tool #7 (Identify Subproblems) then climbs the chain one link at a time, rewriting 6, 7, and finally 8 as a power of 4. Tool #4 (Introduce a Variable) closes it out: once 4^abcd=8, treat the product abcd as a single unknown, match both sides as powers of 2, and solve the little equation that pops out.

1STEP 1

Aim for one combined power

Instead of chasing a, b, c, d one at a time, go after the single power 4^abcd — since (x^a)^b=x^ab, stacked exponents multiply.

(x^a)^b = x^ab
2STEP 2

Climb the chain from 4 up to 8

Raise 4^a=5 to the b, then the c, then the d: each result is the next base, so the chain collapses to 4^abcd=8.

4^a=5, 4^ab=6, 4^abc=7, 4^abcd=8
3STEP 3

Write 4 and 8 as powers of 2

Rewrite both sides in base 2: 4=2² and 8=2³, so 4^abcd=8 becomes 2^(2abcd)=2³.

4=2², 8=2³ → 2²abcd=2³
4STEP 4

Match the exponents and solve

Equal powers of one base force equal exponents, so 2abcd=3, and dividing by 2 gives abcd=3/2 — choice (B).

2 abcd=3 → abcd=3/2=(B)
Answer
3/2
A sanity check on size: each of a,b,c,d is just a bit more than 1 (for instance 4^a=5 makes a slightly above 1), so their product should be a modest number a little above 1 — and 3/2 fits, while 3 (E) would be too big and 1 (A) too small. The deeper check is the chain itself: climbing from base 4 up to 8 means raising 4 to the power 3/2, since the square root of 4 is 2 and 2 cubed is 8. Substituting abcd=3/2 back gives 4^(3/2)=2³=8, matching 7^d=8 exactly.
💡Key takeaway

When equations link up in a chain — each answer becoming the next base — stack them into one power instead of solving each piece, because raising a power to a power just multiplies the exponents.

  • Aim for one combined power
  • Climb the chain from 4 up to 8
  • Write 4 and 8 as powers of 2
  • Match the exponents and solve

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