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question:{0.027^{frac{1}{3}}}+sqrt[4]{{{{16}^3}}}{3^{1}}+{(sqrt{2}1)^0}=______.
answer:To solve the given problem step by step, we start with the original expression: [0.027^{-frac{1}{3}} + sqrt[4]{16^3} - 3^{-1} + (sqrt{2}-1)^0] First, we simplify each term individually: 1. For 0.027^{-frac{1}{3}}, since 0.027 = frac{27}{1000} = (3^3)^{-3} = 3^{-3}, we have: [0.027^{-frac{1}{3}} = (3^{-3})^{-frac{1}{3}} = 3^{(-3) cdot (-frac{1}{3})} = 3^1 = 3] 2. For sqrt[4]{16^3}, recognizing that 16 = 2^4, we get: [sqrt[4]{16^3} = sqrt[4]{(2^4)^3} = 2^{4 cdot frac{3}{4}} = 2^3 = 8] 3. For 3^{-1}, it simplifies directly to: [3^{-1} = frac{1}{3}] 4. For (sqrt{2}-1)^0, any number (except for 0) raised to the power of 0 is 1: [(sqrt{2}-1)^0 = 1] Putting it all together, we have: [0.027^{-frac{1}{3}} + sqrt[4]{16^3} - 3^{-1} + (sqrt{2}-1)^0 = 3 + 8 - frac{1}{3} + 1] Combining like terms and simplifying: [= frac{9}{3} + 8 - frac{1}{3} + 1 = 3 + 8 - frac{1}{3} + 1 = frac{10}{3} + 8 - frac{1}{3} + 1] [= frac{9}{3} + 8 + 1 = 3 + 8 + 1 = 12] Therefore, the final answer is: [boxed{12}]
question:Given that the square roots of a positive number are x+1 and x5, then the number is ____.
answer:To find the positive number whose square roots are given as x+1 and x-5, we proceed as follows: Given that the square roots of the number are x+1 and x-5, we can set up an equation based on the property that the sum of the roots of a quadratic equation equals the negative coefficient of the x term (in this case, we assume the coefficient to be 0 since we are dealing with square roots, not a quadratic equation directly). Therefore, we have: [x + 1 + (x - 5) = 0] Simplifying the equation: [2x - 4 = 0] [2x = 4] [x = 2] Now, substituting x = 2 into one of the given expressions for the square roots, we choose x + 1 for simplicity: [x + 1 = 2 + 1 = 3] Therefore, the positive number is the square of this value: [3^2 = 9] Hence, the positive number is boxed{9}.
question:Let the function f(x) have a derivative f'(x) on mathbb{R}. For any real number x, it satisfies f(x)=4x^2f(x). When x in (infty,0), f'(x)+ frac{1}{2} < 4x. If f(m+1) leqslant f(m)+4m+2, then the range of the real number m is ______. A: [ frac{1}{2},+infty) B: [ frac{3}{2},+infty) C: [1,+infty) D: [2,+infty)
answer:Since f(x)=4x^2-f(-x), it follows that f(x)-2x^2+f(-x)-2x^2=0. Let g(x)=f(x)-2x^2, then we have g(x)+g(-x)=0, which implies that the function g(x) is an odd function. Since for x in (-infty,0), we have f'(x)+ frac{1}{2} < 4x, it follows that g'(x) = f'(x) - 4x < - frac{1}{2}, which means g(x) is decreasing on (-infty,0). As g(x) is an odd function, it will also be decreasing on (0,+infty). If f(m+1) leqslant f(-m)+4m+2, then f(m+1)-2(m+1)^2 leqslant f(-m)-2m^2 + 4m + 2, which simplifies to g(m+1) leqslant g(-m) + 4m+2, but since g(x) is odd, g(-m)=-g(m). So the inequality simplifies further to g(m+1) leqslant -g(m) + 4m + 2, and since g(x) is decreasing in both intervals, g(m+1) < g(m). Therefore, the inequality simplifies to m+1 geqslant -m, which gives us m geqslant -frac{1}{2}. Hence the answer is boxed{A [- frac{1}{2},+infty)}.
question:To perform a brief audit on a certain mall, a method is used to estimate the total monthly sales by checking 2% of the invoices and sales records. The method is as follows: randomly select one out of 25 invoice stubs, such as invoice No.15, and then form an investigation sample with the sales amounts on the invoices sequentially selected after that, such as invoice No.40, 65, 90, and so on. Which sampling method does this represent? A. Simple random sampling B. Systematic sampling C. Stratified sampling D. Other types of sampling
answer:Considering that the population consists of many individuals, a random selection is made from a set of 25 invoice stubs, such as picking invoice No.15. This constitutes the starting point in systematic sampling, where every subsequent 25th iterm following the initial selection, such as invoice No.40, 65, 90, etc., is used to form the survey sample. Therefore, the correct answer is: [ boxed{B.text{ Systematic sampling}} ]