How many of the following are equal to x^x+x^x for all x>0? I: &2x^x II: &x^2x III: &(2x)^x IV: &(2x)^2x

1994 AMC 12 Problem 3

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How many of the following are equal to

x^x+x^x

for all

x\gt0?

\begin{aligned} \mathrm{I:}\quad&2x^x\\ \mathrm{II:}\quad&x^{2x}\\ \mathrm{III:}\quad&(2x)^x\\ \mathrm{IV:}\quad&(2x)^{2x} \end{aligned}

0

1

2

3

4

First combine the two identical terms

Test whether changing the base or exponent preserves

2x^x

for every

x

The given sum is

2x^x,

so expression I is always equal to it. Expression II fails at

x=1,

where its value is

1

instead of

2.

Expression III fails at

x=2,

where its value is

16

instead of

8.

Expression IV fails at

x=1,

where its value is

4

instead of

2.

Hence only one expression works. Thus the correct answer is B.