AMC 10 · 2016 · #2

Grade 8 algebra
Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity
exponentslinear-equations-one-var convert-to-algebra ↑ Prerequisites: exponents
📏 Short solution 💡 2 insights
Problem
Find the value of x that makes 10^x · 100²x=1000⁵ a true equation.

Pick an answer.

(A)
1
(B)
2
(C)
3
(D)
4
(E)
5

AMC 10 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Organize Information in More Ways

The two sides look unmatched because they show three different bases (10, 100, 1000). Tool #15 (Organize Information in More Ways) is the key move: rewrite every number as a power of the single base 10, and the mismatch disappears. Once both sides are a single power of 10, Tool #4 (Introduce a Variable) lets us treat the exponents as a plain equation in x and solve.

1STEP 1

Rewrite as powers of 10

Rewrite with base 10: 100²x=(10²)²x=10⁴x and 1000⁵=(10³)⁵=10¹⁵.

100²x=(10²)²x=10⁴x, 1000⁵=(10³)⁵=10¹⁵
2STEP 2

Combine the left side

Same base, so add exponents: 10^x · 10⁴x=10⁵x, giving 10⁵x=10¹⁵.

10^x · 10⁴x=10^x+4x=10⁵x=10¹⁵
3STEP 3

Match the exponents

Equal powers mean equal exponents: 5x=15, so x=3 — answer (C).

5x=15 → x=15/5=3 → (C)
Answer
3
Check x=3 directly: the left side becomes 10³ · 100⁶=10³ · 10¹²=10¹⁵, which equals 1000⁵=10¹⁵. The two sides match exactly, so (C) is correct. A smaller x like 1 or 2 would leave the left side as 10⁵ or 10¹⁰, both too small to reach 10¹⁵.
💡Key takeaway

When bases look different, turn them all into powers of the same number — then you just match the exponents.

  • Rewrite as powers of 10
  • Combine the left side
  • Match the exponents

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