AMC 10 · 2016 · #2

Grade 8 arithmetic

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)

(B)

(C)

(D)

(E)

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

100²x=(10²)²x=10⁴x, 1000⁵=(10³)⁵=10¹⁵

2STEP 2

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

10^x· 10⁴x=10^x+4x=10⁵x=10¹⁵

3STEP 3

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

More like this