In the equation \(x^7 - 47 = -6\), Shawn's step to multiply both sides of the equation by 7 is actually incorrect for the purpose of simplifying or solving the equation. The equation has a \(x^7\) term, and multiplying both sides by 7 does not directly simplify the original equation or help in isolating \(x\).
The correct reason for manipulation in this case would typically involve adding 47 to both sides to eliminate the -47 term and bring the equation into a more manageable form.
However, if we are to analyze the multiple choice options:
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"Multiplying both sides by 7 undoes the addition in the original problem." - This choice does not apply as the original equation involves subtraction, not addition.
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"Multiplying both sides by 7 undoes the subtraction in the original problem." - This choice might imply an attempt to balance the equation, but it's not accurate in maintaining the integrity of the original problem's operations.
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"Multiplying both sides by 7 undoes the division of x by 7 in the original problem." - This does not apply as there is no division of \(x\) by 7.
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"The negative sign in -6 must be changed to positive." - This is not a valid operation in solving the equation.
Since none of the given responses accurately explain why multiplying both sides by 7 would be a logical or necessary step in solving the original equation, it would be best to clarify that multiplication by 7 is not a valid move in this context.
If he were solving for \(x\), he would likely want to add 47 to both sides first:
\(x^7 = -6 + 47\) leading to \(x^7 = 41\).
In summary, the choice that fits the context the closest in terms of traditional arithmetic operations is:
"Multiplying both sides by 7 undoes the subtraction in the original problem."
But again, note that this does not help in solving the equation correctly.