answer:Let r be the radius of the inscribed circle. Let s be the semiperimeter of the triangle, calculated as s=frac{AB + AC + BC}{2} = frac{6 + 8 + 10}{2} = 12. Let K denote the area of triangle ABC. Using Heron's formula, the area K is: [ K = sqrt{s(s-AB)(s-AC)(s-BC)} = sqrt{12 cdot (12-6) cdot (12-8) cdot (12-10)} = sqrt{12 cdot 6 cdot 4 cdot 2} = sqrt{576} = 24. ] The area of a triangle is also equal to its semiperimeter multiplied by the radius of its inscribed circle: [ K = rs implies 24 = 12r implies r = frac{24}{12} = 2. ] Thus, the radius of the inscribed circle is r=boxed{2}.

answer:To find the center of the image circle, we need to reflect the original center over the line ( y = -x ). Reflection over the line ( y = -x ) involves swapping the ( x ) and ( y ) coordinates of the point and changing the sign of both coordinates. - The original center of the circle is ((8, -3)). - When reflecting the point ((8, -3)) over ( y = -x ), the new coordinates become ((-3, -8)) because both coordinates swap places and both signs are changed. Thus, the new center of the reflected circle is (boxed{(-3, -8)}).

answer:Let's denote the number of times a day Theo eats cookies as ( x ). Theo eats 13 cookies each time he eats cookies. So, in one day, he eats ( 13x ) cookies. He eats cookies 20 days each month, so in one month, he eats ( 20 times 13x = 260x ) cookies. Theo eats 2340 cookies in 3 months, so in one month, he would eat ( frac{2340}{3} = 780 ) cookies. Now we can set up the equation: ( 260x = 780 ) To find ( x ), we divide both sides by 260: ( x = frac{780}{260} ) ( x = 3 ) So, Theo eats cookies boxed{3} times a day.

answer:1. Evaluate each factor: - 81^{1/2} = (9^2)^{1/2} = 9 - 8^{-1/3} = frac{1}{(2^3)^{1/3}} = frac{1}{2} - 32^{1/5} = (2^5)^{1/5} = 2 2. Multiply the results: - 9 cdot frac{1}{2} cdot 2 = 9 The final answer is boxed{9}. Conclusion: The solution is correct as each base was raised to a power, the computation was conducted per exponent laws with reasonable simplifications, and the multiplication of results aligns with the principle of operations on real numbers.