To find expressions equivalent to \( 200(10^x) \), let's first rewrite \( 200 \) in terms of powers of 10:
\[ 200 = 2 \times 100 = 2 \times 10^2 \]
Thus, we can express \( 200(10^x) \) as:
\[ 200(10^x) = (2 \times 10^2)(10^x) = 2(10^2 \times 10^x) = 2(10^{x+2}) \]
This means that \( 200(10^x) \) can be expressed as \( 2(10^{x + 2}) \).
Now let's evaluate the five given options:
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\( 2(10^{2x}) \) - This is not equivalent because \( 2x \) is not the same as \( x + 2 \).
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\( 2(10^{3x}) \) - This is not equivalent since \( 3x \) does not equal \( x + 2 \).
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\( 2(10^{x + 2}) \) - This is equivalent since it matches our derived expression.
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\( 20(10^{2x}) \) - This is not equivalent because \( 20 = 2 \times 10 \), and \( 10^{2x} \) does not match \( 10^{x + 2} \).
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\( 20(10^{x + 1}) \) - This is not equivalent because even though \( 20 = 2 \times 10 \), \( 10^{x + 1} \) does not equal \( 10^{x + 2} \).
Thus, the only expression that is equivalent to \( 200(10^x) \) is:
3. \( 2(10^{x + 2}) \).