2002 AMC 10P Problems/Problem 12

Problem 12

For $f_n(x)=x^n$ and $a \neq 1$ consider

$\text{I. } (f_{11}(a)f_{13}(a))^{14}$

$\text{II. } f_{11}(a)f_{13}(a)f_{14}(a)$

$\text{III. } (f_{11}(f_{13}(a)))^{14}$

$\text{IV. } f_{11}(f_{13}(f_{14}(a)))$

Which of these equal $f_{2002}(a)?$

$\text{(A) I and II only} \qquad \text{(B) II and III only} \qquad \text{(C) III and IV only} \qquad \text{(D) II, III, and IV only} \qquad \text{(E) all of them}$

Solution 1

We can solve this problem with a case by case check of $\text{I., II., III.,}$ and $\text{IV.}$ Since $f_n=x^n,$ $f_{2002}(a)=a^{2002},$ all cases must equal $a^{2002}.$

$\text{I. } (f_{11}(a)f_{13}(a))^{14}$

\begin{align*} (f_{11}(a)f_{13}(a))^{14} \\ &= (a^{11}a^{13})^{14} \\ &= (a^{24})^14 \\ &= a^{336} \\ &\neq a^{2002} \\ \end{align*}

2002 AMC 10P (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
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2002 AMC 10P Problems/Problem 12

Problem 12

Solution 1

See also