2000 AMC 12 Problems/Problem 2

The following problem is from both the 2000 AMC 12 #2 and 2000 AMC 10 #2, so both problems redirect to this page.

Problem

$2000(2000^{2000}) = ?$

$\textbf{(A)} \ 2000^{2001} \qquad \textbf{(B)} \ 4000^{2000} \qquad \textbf{(C)} \ 2000^{4000} \qquad \textbf{(D)} \ 4,000,000^{2000} \qquad \textbf{(E)} \ 2000^{4,000,000}$

Solution

We can use an elementary exponents rule to solve our problem. We know that $a^b\cdot a^c = a^{b+c}$ . Hence, $2000(2000^{2000}) = (2000^{1})(2000^{2000}) = 2000^{2000+1} = 2000^{2001} \Rightarrow \boxed{\textbf{(A) } 2000^{2001}}$

Solution edited by armang32324 and integralarefun

Solution 2

We see that $a(a^{2000})=a^{2001}.$ Only answer choice $\boxed{\textbf{(A)}}$ satisfies this requirement.

-SirAppel

Solution 3

Say you somehow forgot the basic rules of exponents, you could just deduce $2000^{2000}$ is a line of 2000 2000's being multiplied, and if we multiplied by this line by an additional 2000, we would then have $\boxed{\textbf{(A 2001)}}$ 2000's lined up multiplying each other.

itsj

Video Solution (Daily Dose of Math)

https://www.youtube.com/watch?v=h0QtF9J0oPs

~Thesmartgreekmathdude

2000 AMC 12 (Problems • Answer Key • Resources)
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All AMC 12 Problems and Solutions

2000 AMC 10 (Problems • Answer Key • Resources)
Preceded by Problem 1	Followed by Problem 3
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
All AMC 10 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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