AMC 10 · 2008 · #5
Grade 7 algebraFor real numbers a and b, define a∗b=(a−b)2. What is (x−y)2∗(y−x)2?
Pick an answer.
AMC 10 2008 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: A new operation is defined by $a * b=(a-b)^2$: take the two inputs, subtract them, and square the result. Using this rule, find the value of $(x-y)^2 * (y-x)^2$.
Givens: The rule $a * b=(a-b)^2$, which holds for all real numbers $a$ and $b$; The expression to evaluate is $(x-y)^2 * (y-x)^2$, so the left input is $(x-y)^2$ and the right input is $(y-x)^2$; Answer choices: (A) $0$, (B) $x^2+y^2$, (C) $2x^2$, (D) $2y^2$, (E) $4xy$
Unknowns: The single expression that $(x-y)^2 * (y-x)^2$ simplifies to
Understand
Restated: A new operation is defined by $a * b=(a-b)^2$: take the two inputs, subtract them, and square the result. Using this rule, find the value of $(x-y)^2 * (y-x)^2$.
Givens: The rule $a * b=(a-b)^2$, which holds for all real numbers $a$ and $b$; The expression to evaluate is $(x-y)^2 * (y-x)^2$, so the left input is $(x-y)^2$ and the right input is $(y-x)^2$; Answer choices: (A) $0$, (B) $x^2+y^2$, (C) $2x^2$, (D) $2y^2$, (E) $4xy$
Plan
Primary tool: #4 Introduce a Variable
Secondary: #6 Guess and Check, #3 Eliminate Possibilities
The expression looks tangled because the two inputs $(x-y)^2$ and $(y-x)^2$ are themselves squares. Tool #4 (Introduce a Variable) tames this: name the left input $a$ and the right input $b$, so the whole problem becomes the clean $a * b=(a-b)^2$. The key observation then jumps out — $a$ and $b$ are actually the same number — so the difference $a-b$ is $0$. Tool #6 (Guess and Check) confirms the result with concrete numbers, and Tool #3 (Eliminate Possibilities) explains why every non-zero choice (which all still contain $x$ or $y$) must be wrong once the difference collapses to $0$.
Execute — Answer: A
6.EE.A.2 Step 1 Decode the star rule
- The star is not multiplication; it is the defined rule $a * b=(a-b)^2$.
- It says: whatever the two inputs are, subtract the second from the first and square the answer.
- Name the inputs so the structure is visible: let $a=(x-y)^2$ (the left input) and $b=(y-x)^2$ (the right input).
- The task is now to compute $a * b=(a-b)^2$.
💡 Giving the messy inputs short names turns the problem into the simple rule $(a-b)^2$.
7.NS.A.2 Step 2 The two inputs are equal
- Compare $a=(x-y)^2$ with $b=(y-x)^2$.
- The bases $x-y$ and $y-x$ are opposites: $y-x=-(x-y)$.
- Squaring a number and squaring its opposite give the same result, because a negative times a negative is positive: $(-k)^2=k^2$.
- So $(y-x)^2=(x-y)^2$, which means $a$ and $b$ are the exact same number.
💡 Flipping a subtraction only changes its sign, and squaring throws that sign away.
6.EE.A.2 Step 3 Test with real numbers
- Check the equality with easy values.
- Let $x=2$ and $y=1$.
- Then $(x-y)^2=(2-1)^2=1$ and $(y-x)^2=(1-2)^2=(-1)^2=1$ — both inputs are $1$.
- Applying the rule, $1 * 1=(1-1)^2=0$.
- The two inputs really are equal and the star of two equal numbers lands on $0$.
💡 Trying concrete numbers makes the abstract equality something you can see.
6.EE.A.1 Step 4 Subtract equal inputs and square
- Since $a=b$, the difference inside the rule is $a-b=0$, and squaring gives $0^2=0$.
- The variables $x$ and $y$ vanish entirely — the answer does not depend on them.
- Every other choice still contains $x$ or $y$, so they cannot be right when the difference collapses to $0$.
- The value is $0$, which is choice (A).
💡 Subtracting a number from itself always leaves $0$, and $0$ squared is still $0$.
6.EE.A.2 The star is not multiplication; it is the defined rule $a * b=(a-b)^2$. It says: 7.NS.A.2 Compare $a=(x-y)^2$ with $b=(y-x)^2$. The bases $x-y$ and $y-x$ are opposites: $ 6.EE.A.2 Check the equality with easy values. Let $x=2$ and $y=1$. Then $(x-y)^2=(2-1)^2= 6.EE.A.1 Since $a=b$, the difference inside the rule is $a-b=0$, and squaring gives $0^2= Review
Reasonableness: The whole result rests on one fact: the two inputs $(x-y)^2$ and $(y-x)^2$ are identical, so their difference is $0$ and its square is $0$. The numeric test with $x=2,\,y=1$ gave $0$, and a second test — say $x=5,\,y=3$: $(5-3)^2=4$, $(3-5)^2=4$, $4 * 4=(4-4)^2=0$ — gives $0$ again. Because the answer never depends on $x$ or $y$, every choice that still shows an $x$ or $y$ (B, C, D, E) must be a distractor; only the constant (A) $0$ can survive.
Alternative: Skip the variable names and reason directly: the star rule squares the difference of its two inputs, and the two inputs are the same quantity $(x-y)^2$. The difference of a thing with itself is $0$, so the squared difference is $0$ no matter what $x$ and $y$ are — answer (A).
CCSS standards used (min grade 7)
6.EE.A.2Write, read, and evaluate expressions in which letters stand for numbers (Reading the defined rule $a * b=(a-b)^2$, naming the two inputs $a$ and $b$, and substituting values to evaluate it.)7.NS.A.2Apply and extend understanding of multiplication and division of rational numbers (Justifying $(y-x)^2=(x-y)^2$ from the fact that a negative times a negative is positive, so opposite numbers have equal squares.)6.EE.A.1Write and evaluate numerical expressions involving whole-number exponents (Evaluating the final square $0^2=0$ once the difference of the two equal inputs is $0$.)
⭐ When a rule squares the difference of two things and those two things are actually equal, the difference is $0$ — so the whole answer is $0$.
⭐ When a rule squares the difference of two things and those two things are actually equal, the difference is $0$ — so the whole answer is $0$.
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