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

Powers of different bases multiply to another power. Find the exponent.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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 every base over ten

Every base rewrites over ten.

100²x=(10²)²x=10⁴x, 1000⁵=(10³)⁵=10¹⁵

2STEP 2

Multiplying adds the exponents

Multiplying adds the exponents.

10^x · 10⁴x=10^x+4x=10⁵x=10¹⁵

3STEP 3

Match the exponents

Matching exponents gives 3, choice (C).

5x=15 → x=15/5=3 → (C)

Answer

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 every base over ten

Multiplying adds the exponents

Match the exponents

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