answer:If g(x) is an odd function, this means that g(-x) = -g(x). We need to evaluate g(g(g(g(x)))) under the assumption that g(x) is odd. 1. Apply the property of odd function to g(g(x)): [ g(g(-x)) = g(-g(x)) = -g(g(x)). ] Thus, g(g(x)) is an even function since replacing x with -x results in the same function value but negated. 2. Now, evaluate g(g(g(g(x)))) keeping in mind that g(g(x)) is even: [ g(g(g(g(-x)))) = g(g(g(g(x)))) ] because g(g(x)) being even implies that g applied to g(x) or -g(x) yields the same result. Therefore, g(g(g(g(x)))) is an even function. Conclusion: The function g(g(g(g(x)))) is boxed{text{even}}.

answer:Let's denote the percentage of lemonade in the first solution as L%. The percentage of carbonated water in the first solution is then 100% - L%. The second solution is 45% lemonade and 55% carbonated water. We are mixing these two solutions such that 67.99999999999997% (which we can round to 68% for simplicity) of the mixture is the first solution, and the remaining 32% is the second solution. The total percentage of carbonated water in the mixture is 72%. We can set up an equation to represent the mixture of carbonated water from both solutions: 68% * (100% - L%) + 32% * 55% = 72% Now, let's solve for L: 0.68 * (100 - L) + 0.32 * 55 = 72 68 - 0.68L + 17.6 = 72 68 - 0.68L = 72 - 17.6 68 - 0.68L = 54.4 0.68L = 68 - 54.4 0.68L = 13.6 L = 13.6 / 0.68 L = 20 Therefore, the first solution is boxed{20%} lemonade.

answer:To form a 3-digit even number using the digits 0, 1, 2, 3, 4, 5, we must consider different cases based on the number of even digits selected and their positions: 1. **One even digit**: If only one even digit is selected, it must be in the units place to ensure the number is even. The number of ways to choose the even digit is C_{3}^{1}, and the number of ways to arrange the remaining two digits is A_{3}^{2}. Therefore, the total is C_{3}^{1} times A_{3}^{2} = 3 times 6 = 18. 2. **Two even digits, including zero**: If one of the even digits is zero, we have two subcases: - **Zero in the units place**: This is not possible because the number would not be even. - **Zero in the tens place**: The number of ways to choose the non-zero even digit for the units place is C_{2}^{1}, and the choice for the hundreds place is C_{3}^{1}. Thus, the total is C_{2}^{1} times C_{3}^{1} = 2 times 3 = 6. 3. **Two even digits, excluding zero**: If neither of the even digits is zero, the even digit for the units place can be chosen in C_{2}^{1} ways. There are A_{2}^{1} choices for the hundreds place and A_{2}^{2} ways to arrange the remaining digit. The total is C_{3}^{1} times A_{2}^{1} times A_{2}^{2} = 3 times 2 times 2 = 12. 4. **Three even digits**: If all three selected digits are even, the hundred's place can't be zero, so there are A_{2}^{1} ways to choose the hundreds place and the remaining two digits can be arranged in A_{2}^{2} ways. Therefore, the total number of combinations is A_{2}^{1} times A_{2}^{2} = 2 times 2 = 4. Adding all the cases together: 18 + 6 + 12 + 4 = 40 However, upon reviewing the combinations, it seems there was a miscalculation in the initial translation regarding the subcase with zero in a position other than the units place. The correct calculation for two even digits, including zero, without zero being in the units place should be as follows: - **Zero not in the units place**: Choose the position for zero and the hundred's digit from the 2 non-zero even digits in C_{2}^{1} ways, then choose the remaining odd digit in C_{3}^{1} ways. This results in C_{2}^{1} times C_{3}^{1} = 2 times 3 = 6. However, since zero cannot be in the units place, if zero is in the hundred's place, we can only place the other even number in the tens place and have 3 choices for the units place. This gives us 1 times C_{3}^{1} = 3. Thus, the revised correct total of combinations is: 18 + 3 + 12 + 4 = boxed{37}

answer:The expression given is sqrt{50y} cdot sqrt{5y} cdot sqrt{45y}. 1. **Combine and simplify the expression**: Combine all the terms under a single radical: [ sqrt{50y} cdot sqrt{5y} cdot sqrt{45y} = sqrt{(50y)(5y)(45y)} ] 2. **Calculate the product inside the radical**: Multiply the numbers and the variable terms: [ (50y)(5y)(45y) = 50 cdot 5 cdot 45 cdot y^3 = 11250y^3 ] 3. **Factorize and simplify**: - Prime factorization of 11250: (11250 = 2 cdot 5^3 cdot 3^2) - This factorization can be rewritten in terms of square roots: [ sqrt{11250y^3} = sqrt{2 cdot 5^3 cdot 3^2 cdot y^3} ] - Recognize perfect squares and simplify: [ sqrt{2 cdot (5^2 cdot 3^2 cdot y^2) cdot 5 cdot y} = 15y sqrt{10y} ] 4. **Final simplification**: The expression simplifies to: [ boxed{15y sqrt{10y}} ]