AMC 10 · 2005 · #17
Grade 8 arithmeticSuppose that 4a=5, 5b=6, 6c=7, and 7d=8. What is a⋅b⋅c⋅d?
Pick an answer.
AMC 10 2005 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: Four numbers $a,b,c,d$ are hidden inside four equations: $4^a=5$, $5^b=6$, $6^c=7$, and $7^d=8$. Find the value of the product $a\cdot b\cdot c\cdot d$.
Givens: $4^a=5$; $5^b=6$; $6^c=7$; $7^d=8$; Answer choices: (A) $1$, (B) $\frac{3}{2}$, (C) $2$, (D) $\frac{5}{2}$, (E) $3$
Unknowns: The value of the product $a\cdot b\cdot c\cdot d$
Understand
Restated: Four numbers $a,b,c,d$ are hidden inside four equations: $4^a=5$, $5^b=6$, $6^c=7$, and $7^d=8$. Find the value of the product $a\cdot b\cdot c\cdot d$.
Givens: $4^a=5$; $5^b=6$; $6^c=7$; $7^d=8$; Answer choices: (A) $1$, (B) $\frac{3}{2}$, (C) $2$, (D) $\frac{5}{2}$, (E) $3$
Plan
Primary tool: #16 Change Focus / Count the Complement
Secondary: #7 Identify Subproblems, #4 Introduce a Variable
Chasing $a,b,c,d$ separately is hopeless because none of them is a nice number. Tool #16 (Change Focus) is the key move: stop asking for each exponent and instead ask what a single power, $4^{a b c d}$, equals — because the rule $(x^a)^b=x^{ab}$ means stacking the four exponents is the same as multiplying them. Tool #7 (Identify Subproblems) then climbs the chain one link at a time, rewriting $6$, $7$, and finally $8$ as a power of $4$. Tool #4 (Introduce a Variable) closes it out: once $4^{abcd}=8$, treat the product $abcd$ as a single unknown, match both sides as powers of $2$, and solve the little equation that pops out.
Execute — Answer: B
8.EE.A.1 Step 1 Aim for one combined power
- Instead of finding $a,b,c,d$ one by one, focus on the single quantity $4^{a b c d}$.
- The power-of-a-power rule says $(x^a)^b=x^{ab}$: raising to one exponent and then another is the same as multiplying the exponents.
- So if we can build $4$ raised to the product $a b c d$ and see what number it equals, we can read the product straight off.
💡 Multiplying the four exponents is exactly what happens when you stack the four powers on top of each other.
8.EE.A.1 Step 2 Climb the chain from 4 up to 8
- Start with $4^a=5$.
- Now raise both sides to the $b$: $5^b=(4^a)^b=4^{ab}$, and the givens say $5^b=6$, so $4^{ab}=6$.
- Raise to the $c$: $6^c=(4^{ab})^c=4^{abc}=7$.
- Raise to the $d$: $7^d=(4^{abc})^d=4^{abcd}=8$.
- Each link reuses the previous result as its base, and the exponents pile up by multiplication, so the whole chain collapses to $4^{abcd}=8$.
💡 Because the result of each equation is the base of the next, the four steps telescope into one clean statement about $4$ raised to the product.
6.EE.A.1 Step 3 Write 4 and 8 as powers of 2
- Both sides of $4^{abcd}=8$ can be written with base $2$.
- Since $4=2^2$ and $8=2^3$, the left side becomes $(2^2)^{abcd}=2^{2abcd}$ and the right side becomes $2^3$.
- Now the equation reads $2^{2abcd}=2^{3}$.
💡 Putting both sides over the same base $2$ lets you compare the two towers just by looking at their exponents.
8.EE.A.1 Step 4 Match the exponents and solve
- When two powers of the same base are equal, their exponents must be equal, so $2\,abcd = 3$.
- Treat the product $abcd$ as a single unknown and divide both sides by $2$: $abcd=\frac{3}{2}$.
- So the answer is $\textbf{(B)}\ \frac{3}{2}$.
💡 Equal powers of the same base force equal exponents, turning the whole problem into a one-step equation.
8.EE.A.1 Instead of finding $a,b,c,d$ one by one, focus on the single quantity $4^{a b c 8.EE.A.1 Start with $4^a=5$. Now raise both sides to the $b$: $5^b=(4^a)^b=4^{ab}$, and t 6.EE.A.1 Both sides of $4^{abcd}=8$ can be written with base $2$. Since $4=2^2$ and $8=2^ 8.EE.A.1 When two powers of the same base are equal, their exponents must be equal, so $2 Review
Reasonableness: A sanity check on size: each of $a,b,c,d$ is just a bit more than $1$ (for instance $4^a=5$ makes $a$ slightly above $1$), so their product should be a modest number a little above $1$ — and $\frac{3}{2}$ fits, while $3$ (E) would be too big and $1$ (A) too small. The deeper check is the chain itself: going from base $4$ up to $8$ means multiplying by $2$ exactly $\frac{3}{2}$ times when each 'times' is measured against base $4$, since $4^{3/2}=8$. Substituting $abcd=\frac{3}{2}$ back gives $4^{3/2}=(\sqrt{4})^3=2^3=8$, matching $7^d=8$ exactly.
Alternative: Read each exponent as a change-of-base ratio: $a=\dfrac{\ln 5}{\ln 4}$, $b=\dfrac{\ln 6}{\ln 5}$, $c=\dfrac{\ln 7}{\ln 6}$, $d=\dfrac{\ln 8}{\ln 7}$. Multiplying, every numerator cancels the next denominator and the product telescopes to $\dfrac{\ln 8}{\ln 4}=\dfrac{3\ln 2}{2\ln 2}=\dfrac{3}{2}$ — the same answer (B).
CCSS standards used (min grade 8)
8.EE.A.1Know and apply the properties of integer exponents (Using the power-of-a-power rule $(x^a)^b=x^{ab}$ to chain the four equations into $4^{abcd}=8$, and equating exponents once both sides share base $2$ to get $2\,abcd=3$.)6.EE.A.1Write and evaluate numerical expressions involving whole-number exponents (Recognizing $4=2^2$ and $8=2^3$ so both sides of $4^{abcd}=8$ can be rewritten over the common base $2$.)
⭐ When equations link up in a chain — each answer becoming the next base — stack them into one power instead of solving each piece, because raising a power to a power just multiplies the exponents.
⭐ When equations link up in a chain — each answer becoming the next base — stack them into one power instead of solving each piece, because raising a power to a power just multiplies the exponents.
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