answer:To find the power of 11 in the expression, we need to understand how the total number of prime factors is calculated. The expression given is (11)^n * (4)^11 * (7)^5, where n is the power of 11 we need to find. First, let's break down the expression into its prime factors: - (11)^n will contribute n prime factors, all of which are 11. - (4)^11 is the same as (2^2)^11, which is (2)^22. So, this contributes 22 prime factors of 2. - (7)^5 contributes 5 prime factors, all of which are 7. We are given that the total number of prime factors is 29. So, we can set up the following equation: n (from 11^n) + 22 (from 4^11) + 5 (from 7^5) = 29 Now, we can solve for n: n + 22 + 5 = 29 n + 27 = 29 n = 29 - 27 n = 2 Therefore, the power of 11 in the expression is boxed{2} .

answer:Consider A and B as fixed points in the plane. The set of possible locations of point C is the circle centered at B with radius 12. To maximize angle A, overline{AC} must be tangent to the circle at point C. This condition creates a right triangle since angle C = 90^circ. Applying the Pythagorean theorem in triangle ABC, where overline{AC} is the hypotenuse: [ AC = sqrt{AB^2 - BC^2} = sqrt{18^2 - 12^2} = sqrt{324 - 144} = sqrt{180} = 6sqrt{5}. ] Now, calculate tan A: [ tan A = frac{text{opposite side (BC)}}{text{adjacent side (part of AC)}} = frac{12}{6sqrt{5}} = frac{2sqrt{5}}{5}. ] Thus, the largest possible value of tan A is boxed{frac{2sqrt{5}}{5}}.

answer:(1) When a=-frac{3}{2}, the function becomes f(x)=frac{1}{3}x^{3}-frac{3}{2}x^{2}+2x-frac{3}{2}. The derivative of the function is f'(x)=x^{2}-3x+2. The slope of the tangent line at the point (3,f(3)) is given by f'(3)=9-9+2=2. The point of tangency is (3,0). Hence, the equation of the tangent line is y=2(x-3), or equivalently, boxed{2x-y-6=0}. (2) The derivative of the function f(x)=frac{1}{3}x^{3}-frac{3}{2}x^{2}+2x+a is f'(x)=x^{2}-3x+2. When 1<x<2, f'(x)<0 and f(x) is decreasing; when x>2 or x<1, f'(x)>0 and f(x) is increasing. The function f(x) attains its maximum value at x=1, which is frac{5}{6}+a, and its minimum value at x=2, which is frac{2}{3}+a. Since the equation f(x)=0 has three distinct real roots, we have frac{5}{6}+a>0 and frac{2}{3}+a<0. Solving these inequalities, we get -frac{5}{6}<a<-frac{2}{3}. Therefore, the range of values for a is boxed{(-frac{5}{6},-frac{2}{3})}.

answer:Let's denote the number of female employees as F. According to the problem, if 3 more men were hired, the number of male employees would be 189 + 3 = 192. At this point, the ratio of male to female employees would be 8 : 9. This means that for every 8 male employees, there are 9 female employees. So we can write the following equation: 192 / F = 8 / 9 Now, we can solve for F: 192 * 9 = 8 * F 1728 = 8F F = 1728 / 8 F = 216 So there are currently 216 female employees in the company. Now, we can find the current ratio of male to female employees using the original number of male employees (189) and the number of female employees we just calculated (216): Male : Female = 189 : 216 To simplify this ratio, we can divide both numbers by their greatest common divisor. In this case, the greatest common divisor of 189 and 216 is 27. 189 / 27 = 7 216 / 27 = 8 So the current ratio of male to female employees in the company is boxed{7} : 8.