3 Perfoun the operation sand operatorname(lif)_(2) (x^2-81)/(x^2)+18 times 81 div ((x-3)(x+1))/(x^2)-6 x+9

To perform the operation \( \operatorname{lif}_{2} \frac{x^{2}-81}{x^{2}+18 \times 81} \div \frac{(x-3)(x+1)}{x^{2}-6 x+9} \), we need to follow the steps below:<br /><br />Step 1: Simplify the expression inside the fraction.<br />The numerator \( x^{2}-81 \) can be factored as \( (x-9)(x+9) \).<br />The denominator \( x^{2}+18 \times 81 \) can be simplified as \( x^{2}+1458 \).<br /><br />So, the expression becomes \( \frac{(x-9)(x+9)}{x^{2}+1458} \).<br /><br />Step 2: Simplify the denominator of the second fraction.<br />The denominator \( x^{2}-6x+9 \) can be factored as \( (x-3)^{2} \).<br /><br />So, the expression becomes \( \frac{(x-3)(x+1)}{(x-3)^{2}} \).<br /><br />Step 3: Rewrite the division as multiplication by the reciprocal.<br />The expression becomes \( \frac{(x-9)(x+9)}{x^{2}+1458} \times \frac{(x-3)^{2}}{(x-3)(x+1)} \).<br /><br />Step 4: Cancel out common factors.<br />The factor \( (x-3) \) in the numerator and denominator can be canceled out.<br /><br />So, the expression becomes \( \frac{(x-9)(x+9)}{x^{2}+1458} \times \frac{(x-3)}{(x+1)} \).<br /><br />Step 5: Multiply the fractions.<br />The expression becomes \( \frac{(x-9)(x+9)(x-3)}{(x^{2}+1458)(x+1)} \).<br /><br />Therefore, the final answer is \( \frac{(x-9)(x+9)(x-3)}{(x^{2}+1458)(x+1)} \).