AMC 10 · 2003 · #9
Grade 8 algebraFind the value of x that satisfies the equation
25−2=526/x⋅2517/x548/x.
Pick an answer.
AMC 10 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: An equation says that $25^{-2}$ equals the fraction $\dfrac{5^{48/x}}{5^{26/x}\cdot 25^{17/x}}$. Every number in it is a power of $5$ or a power of $25$, and the unknown $x$ sits in the exponents. Find the value of $x$ that makes the equation true.
Givens: The equation $25^{-2}=\dfrac{5^{48/x}}{5^{26/x}\cdot 25^{17/x}}$; $25=5^2$, so every base can be written as a power of $5$; Answer choices: (A) $2$, (B) $3$, (C) $5$, (D) $6$, (E) $9$
Unknowns: The value of $x$ that satisfies the equation
Understand
Restated: An equation says that $25^{-2}$ equals the fraction $\dfrac{5^{48/x}}{5^{26/x}\cdot 25^{17/x}}$. Every number in it is a power of $5$ or a power of $25$, and the unknown $x$ sits in the exponents. Find the value of $x$ that makes the equation true.
Givens: The equation $25^{-2}=\dfrac{5^{48/x}}{5^{26/x}\cdot 25^{17/x}}$; $25=5^2$, so every base can be written as a power of $5$; Answer choices: (A) $2$, (B) $3$, (C) $5$, (D) $6$, (E) $9$
Plan
Primary tool: #15 Organize Information in More Ways
Secondary: #7 Identify Subproblems, #6 Guess and Check
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.
Execute — Answer: B
8.EE.A.1 Step 1 Write every base as a power of 5
- The only obstacle is the two different bases, $5$ and $25$.
- Since $25=5^2$, replace each $25$ with $5^2$.
- The left side becomes $25^{-2}=(5^2)^{-2}=5^{-4}$, using the rule that a power raised to a power multiplies the exponents.
- In the denominator, $25^{17/x}=(5^2)^{17/x}=5^{34/x}$ the same way.
💡 When numbers share a hidden common base, rewriting them that way turns a mixed expression into one you can combine.
8.EE.A.1 Step 2 Combine the denominator
- The denominator is now $5^{26/x}\cdot 5^{34/x}$.
- Multiplying powers of the same base adds the exponents, so $5^{26/x}\cdot 5^{34/x}=5^{(26+34)/x}=5^{60/x}$.
- The right side is now a single power of $5$ over a single power of $5$.
💡 Same base times same base just piles the exponents together into one.
8.EE.A.1 Step 3 Divide to get one power
- Now the right side is $\dfrac{5^{48/x}}{5^{60/x}}$.
- Dividing powers of the same base subtracts the exponents, so this equals $5^{48/x-60/x}=5^{-12/x}$.
- The equation has become $5^{-4}=5^{-12/x}$.
💡 Dividing same-base powers cancels the shared factors, leaving the difference of the exponents.
6.EE.B.7 Step 4 Match exponents and solve
- Both sides are now the same base $5$, and they are equal, so their exponents must be equal: $-4=-\dfrac{12}{x}$.
- Multiply both sides by $x$ to get $-4x=-12$, then divide by $-4$ to find $x=3$.
- That is choice (B).
💡 If two equal numbers are the same base raised to a power, the powers themselves have to match.
8.EE.A.1 The only obstacle is the two different bases, $5$ and $25$. Since $25=5^2$, repl 8.EE.A.1 The denominator is now $5^{26/x}\cdot 5^{34/x}$. Multiplying powers of the same 8.EE.A.1 Now the right side is $\dfrac{5^{48/x}}{5^{60/x}}$. Dividing powers of the same 6.EE.B.7 Both sides are now the same base $5$, and they are equal, so their exponents mus Review
Reasonableness: Put $x=3$ back into the original. The exponents become $\tfrac{48}{3}=16$, $\tfrac{26}{3}$, and $\tfrac{17}{3}$. The right side is $\dfrac{5^{16}}{5^{26/3}\cdot 5^{34/3}}=\dfrac{5^{16}}{5^{60/3}}=\dfrac{5^{16}}{5^{20}}=5^{-4}$, and the left side is $25^{-2}=5^{-4}$. Both sides match, so $x=3$ is correct. A common trap is mishandling $25^{17/x}$ as $5^{17/x}$ 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.
Alternative: Tool #6 (Guess and Check): since the answer is one of five choices, test them directly. The combined right side is $5^{-12/x}$ and the left is $5^{-4}$, so you need $-\tfrac{12}{x}=-4$, i.e. $\tfrac{12}{x}=4$. Only $x=3$ makes $\tfrac{12}{x}$ equal to $4$; $x=2$ gives $6$, $x=6$ gives $2$, so those fail. This lands on (B) without solving an equation formally.
CCSS standards used (min grade 8)
8.EE.A.1Know and apply the properties of integer exponents (Rewriting $25$ as $5^2$, then using the power-of-a-power, product, and quotient rules to collapse each side into a single power of $5$.)6.EE.B.7Solve real-world problems by writing and solving equations of the form px = q (Turning the matched exponents $-4=-\tfrac{12}{x}$ into $-4x=-12$ and solving to get $x=3$.)
⭐ 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.
⭐ 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.
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