Subjects algebra

Expression Simplification D1645D

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1. **State the problem:** Simplify the expression $$ (1 - x)(\sqrt{1 - x^2})(\sqrt{1 - x^2}) (2 + 1)(\sqrt{1 - x^2}) $$. 2. **Rewrite the expression:** Note that $$ (\sqrt{1 - x^2})(\sqrt{1 - x^2}) = 1 - x^2 $$ because $$ \sqrt{a} \times \sqrt{a} = a $$ for nonnegative $$a$$. So the expression becomes: $$ (1 - x)(1 - x^2)(3)(\sqrt{1 - x^2}) $$ 3. **Combine constants:** $$ 2 + 1 = 3 $$, so we have: $$ 3(1 - x)(1 - x^2)(\sqrt{1 - x^2}) $$ 4. **Simplify the product:** The expression is: $$ 3(1 - x)(1 - x^2)(\sqrt{1 - x^2}) $$ 5. **Rewrite $$ (1 - x^2)(\sqrt{1 - x^2}) $$ as $$ (1 - x^2)^{1 + \frac{1}{2}} = (1 - x^2)^{\frac{3}{2}} $$. 6. **Final simplified form:** $$ 3(1 - x)(1 - x^2)^{\frac{3}{2}} $$ **Answer:** $$ \boxed{3(1 - x)(1 - x^2)^{\frac{3}{2}}} $$