1. Stating the problem: Evaluate the expression $\frac{\sqrt{(32)(165)(37)}}{\sqrt{25}\,(46)(53)} - 5\sqrt{(34)(37)}$. 2. Formula used: Use the product rule for square roots $\sqrt{ab}=\sqrt{a}\sqrt{b}$ and extract perfect squares to simplify radicals. 3. Compute the numerator product: $(32)(165)(37)=195360$ so $\sqrt{(32)(165)(37)}=\sqrt{195360}$. 4. Prime factorize 195360: $195360=2^5\cdot3\cdot5\cdot11\cdot37$ so $\sqrt{195360}=2^{2}\sqrt{2\cdot3\cdot5\cdot11\cdot37}=4\sqrt{12210}$. 5. Compute the denominator: $\sqrt{25}(46)(53)=5\cdot46\cdot53=12190$. 6. First term simplifies to $\dfrac{4\sqrt{12210}}{12190}=\dfrac{2\sqrt{12210}}{6095}$ after dividing numerator and denominator by 2. 7. Second term simplifies: $5\sqrt{(34)(37)}=5\sqrt{1258}$ and $1258=2\cdot17\cdot37$ has no square factors to extract. 8. Factor the common $\sqrt{37}$: since $\sqrt{12210}=\sqrt{37}\sqrt{330}$ and $\sqrt{1258}=\sqrt{37}\sqrt{34}$, the expression becomes $\sqrt{37}\left(\dfrac{2\sqrt{330}}{6095}-5\sqrt{34}\right)$. 9. Decimal approximation: $\dfrac{2\sqrt{12210}}{6095}\approx0.03626$ and $5\sqrt{1258}\approx177.34$ so the value is approximately $0.03626-177.34\approx-177.304$. 10. Final exact simplified form: $\displaystyle\sqrt{37}\left(\dfrac{2\sqrt{330}}{6095}-5\sqrt{34}\right)$. 11. Final numeric approximation: $\approx-177.304$.