To square the expression \( (9\sqrt{91} - 8\sqrt{14})^2 \), we can use the formula for the square of a binomial:
\[ (a - b)^2 = a^2 - 2ab + b^2 \]
Here, let \( a = 9\sqrt{91} \) and \( b = 8\sqrt{14} \).
Now, let's compute each term:
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Calculate \( a^2 \): \[ a^2 = (9\sqrt{91})^2 = 81 \cdot 91 = 7371 \]
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Calculate \( b^2 \): \[ b^2 = (8\sqrt{14})^2 = 64 \cdot 14 = 896 \]
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Calculate \( 2ab \): \[ 2ab = 2(9\sqrt{91})(8\sqrt{14}) = 144\sqrt{1274} \] To find \( \sqrt{1274} \): \[ 1274 = 91 \cdot 14 \] So: \[ 2ab = 144\sqrt{1274} \]
Now putting it all together: \[ (9\sqrt{91} - 8\sqrt{14})^2 = a^2 - 2ab + b^2 \] \[ = 7371 - 144\sqrt{1274} + 896 \]
Combine \( a^2 \) and \( b^2 \): \[ 7371 + 896 = 8267 \]
Thus, the final form is: \[ (9\sqrt{91} - 8\sqrt{14})^2 = 8267 - 144\sqrt{1274} \]
In simplest form, the answer is: \[ \boxed{8267 - 144\sqrt{1274}} \]