AMC 10 · 2003 · #9

Grade 8 algebra

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

exponentslinear-equations-one-var identify-subproblems ↑ Prerequisites: exponents

📏 Medium solution 💡 1 insight

Problem

The equation

25^{-2} = \frac{5^{48/x}}{5^{26/x} \cdot 25^{17/x}}

holds for exactly one value of the unknown, where every base is 5 or 25 and

x

appears only inside the exponents. Find that value of

x

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Organize Information in More Ways

The equation looks tangled only because it mixes base 5 and base 25. Tool #15 (Organize Information in More Ways) is the spine: rewrite every term as a power of the single base 5, and the whole expression collapses into one power of 5 on each side. Tool #7 (Identify Subproblems) keeps that tidy by handling one piece at a time — first the left side, then the denominator, then the combined right side. Once both sides are powers of the same base, equal bases mean equal exponents, leaving a tiny equation to solve. Tool #6 (Guess and Check) is the natural safety net for a multiple-choice answer: any candidate can be dropped straight back into the original to confirm it.

1STEP 1

Write every base as a power of 5

Bases 5 and 25 are mixed, but $25=5^2$ : the left side is $25^{-2}=5^{-4}$ , and the denominator's $25^{17/x}$ becomes $5^{34/x}$ .

25⁻²=5⁻⁴ 25¹7/x=5³4/x

2STEP 2

Combine the denominator

Multiplying same-base powers adds exponents, so $5^{26/x} \cdot 5^{34/x}$ collapses to $5^{60/x}$ .

5²6/x · 5³4/x=5⁶0/x

3STEP 3

Divide to get one power

Dividing same-base powers subtracts exponents, so the right side $\frac{5^{48/x}}{5^{60/x}}$ is $5^{-12/x}$ , and the equation is $5^{-4}=5^{-12/x}$ .

(5⁴8/x)/(5⁶0/x)=5^(48-60)/x=5⁻¹2/x

4STEP 4

Match exponents and solve

Same base and equal values force equal exponents: $-4=-\frac{12}{x}$ , so $-4x=-12$ and $x=3$ , choice (B).

-4=-12/x → -4x=-12 → x=3 → (B)

Answer

Put

x=3

back into the original. The exponents become

\frac{48}{3}=16

\frac{26}{3}

, and

\frac{17}{3}

. The right side is

\frac{5^{16}}{5^{26/3} \cdot 5^{34/3}} = \frac{5^{16}}{5^{20}} = 5^{-4}

, and the left side is

25^{-2}=5^{-4}

. Both sides match, so

x=3

is correct. A common trap is mishandling

25^{17/x}

5^{17/x}

instead of

5^{34/x}

; that would give

5^{-4}=5^{5/x}

and a negative

x

, which is not among the answers — a signal that the

25=5^2

conversion was dropped.

💡Key takeaway

When an exponent equation mixes bases like 5 and 25, rewrite everything as one base, combine into a single power on each side, then just match the exponents.

Write every base as a power of 5
Combine the denominator
Divide to get one power
Match exponents and solve

Write every base as a power of 5

Combine the denominator

Divide to get one power

Match exponents and solve

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