To find the width of the table top, we can use the formula for the area of a rectangle, which is given by:
\[ \text{Area} = \text{length} \times \text{width} \]
We are given the area of the rectangle as \( 3^7 \) square inches and the length as \( 3^4 \) inches. We can let the width be \( w \).
Substituting the known values into the area formula, we have:
\[ 3^7 = 3^4 \times w \]
To solve for \( w \), we can divide both sides of the equation by \( 3^4 \):
\[ w = \frac{3^7}{3^4} \]
Using the properties of exponents, specifically that \( \frac{a^m}{a^n} = a^{m-n} \), we can simplify the right side:
\[ w = 3^{7-4} = 3^3 \]
We can further calculate \( 3^3 \):
\[ 3^3 = 27 \]
Therefore, the width of the table top is:
\[ \boxed{27} \text{ inches} \]
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