To find the product of \( 7 \frac{1}{2} \) and \( 10 \frac{1}{2} \) using the difference of squares identity, we start by converting the mixed numbers into improper fractions:
\[ 7 \frac{1}{2} = \frac{15}{2} \quad \text{and} \quad 10 \frac{1}{2} = \frac{21}{2} \]
Now we will calculate the product:
\[ \left( \frac{15}{2} \right) \left( \frac{21}{2} \right) = \frac{15 \times 21}{2 \times 2} = \frac{315}{4} \]
To express this as a mixed number, we can divide \( 315 \) by \( 4 \):
\[ 315 \div 4 = 78 \quad \text{remainder} \quad 3 \]
So,
\[ \frac{315}{4} = 78 \frac{3}{4} \]
Thus, the product of \( 7 \frac{1}{2} \) and \( 10 \frac{1}{2} \) is:
\[ \boxed{78 \frac{3}{4}} \]
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