Step 1: Simplify the expression $(1+\sqrt{3})(1-\sqrt{3})$. This expression is in the form $(a+b)(a-b)$, which simplifies to $a^2 - b^2$. Here, $a=1$ and $b=\sqrt{3}$. $$ (1+\sqrt{3})(1-\sqrt{3}) = 1^2 - (\sqrt{3})^2 $$ $$ = 1 - 3 $$ $$ = -2 $$ The simplified expression is $\boxed{-2}$. Step 2: Evaluate $\frac{1}{1+\sqrt{3}}$ using the result from Step 1. To evaluate $\frac{1}{1+\sqrt{3}}$, we can rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is $(1-\sqrt{3})$. $$ \frac{1}{1+\sqrt{3}} = \frac{1}{1+\sqrt{3}} \times \frac{1-\sqrt{3}}{1-\sqrt{3}} $$ Using the result from Step 1, we know that $(1+\sqrt{3})(1-\sqrt{3}) = -2$. $$ = \frac{1-\sqrt{3}}{-2} $$ $$ = \frac{-( \sqrt{3}-1)}{-2} $$ $$ = \frac{\sqrt{3}-1}{2} $$ Step 3: Substitute the given value of $\sqrt{3} = 1.7321$. $$ \frac{\sqrt{3}-1}{2} = \frac{1.7321 - 1}{2} $$ $$ = \frac{0.7321}{2} $$ $$ = 0.36605 $$ Step 4: Round the result to 3 significant figures. The first three significant figures are 3, 6, and 6. The fourth digit is 0, so we do not round up. $$ 0.36605 \approx 0.366 $$ The value of $\frac{1}{1+\sqrt{3}}$ to 3 significant figures is $\boxed{0.366}$. That's 2 down. 3 left today — send the next one.