answer:To prove the requested inequalities, we use properties derived from the parabolic curve and concepts about the center of mass and projections. 1. First, consider the parabola (y = x^2) for (x geq 0). Take points (A_1, A_2, ldots, A_n) on the parabola with abscissas (x_1, x_2, ldots, x_n). 2. Let point (Z) have coordinates (left(x_0, y_0right)), which is the center of mass of the material points (left(A_1, m_1right), left(A_2, m_2right), ldots, left(A_n, m_nright)). 3. Let (X) be the projection of the point (Z) on the (x)-axis and (Q) be the intersection of the line (XZ) with the parabola. From theorem regarding distances in parabolas, we know that: 4. (XQ < XZ). Now, calculate these distances using the given coordinates and projections: 5. The vertical distance (XZ) is (y_0), defined as: [ y_0 = frac{m_1 y_1 + m_2 y_2 + cdots + m_n y_n}{M} = frac{m_1 x_1^2 + m_2 x_2^2 + cdots + m_n x_n^2}{M}, ] where (M = m_1 + m_2 + cdots + m_n). 6. The vertical distance (XQ) corresponds to: [ x_0^2 = left(frac{m_1 x_1 + m_2 x_2 + cdots + m_n x_n}{M}right)^2. ] 7. Given that (XQ < XZ), we thus have: [ left(frac{m_1 x_1 + m_2 x_2 + cdots + m_n x_n}{M}right)^2 < frac{m_1 x_1^2 + m_2 x_2^2 + cdots + m_n x_n^2}{M}. ] Taking the square root of both sides, we get: [ sqrt{frac{m_1 x_1^2 + m_2 x_2^2 + cdots + m_n x_n^2}{M}} > frac{m_1 x_1 + m_2 x_2 + cdots + m_n x_n}{M}. ] This completes the proof for the weighted inequality. 8. Now, if we set (m_1 = m_2 = ldots = m_n = 1), the expression simplifies to the unweighted form: [ left(frac{x_1 + x_2 + cdots + x_n}{n}right)^2 < frac{x_1^2 + x_2^2 + cdots + x_n^2}{n}. ] Taking the square root of both sides, we get: [ sqrt{frac{x_1^2 + x_2^2 + cdots + x_n^2}{n}} > frac{x_1 + x_2 + cdots + x_n}{n}. ] This completes the proof of the unweighted inequality. Conclusion: [ boxed{sqrt{frac{x_1^2 + x_2^2 + cdots + x_n^2}{n}} > frac{x_1 + x_2 + cdots + x_n}{n}} ] and [ boxed{sqrt{frac{m_1 x_1^2 + m_2 x_2^2 + cdots + m_n x_n^2}{M}} > frac{m_1 x_1 + m_2 x_2 + cdots + m_n x_n}{M}}. ] These inequalities were demonstrated using properties of parabolas and the geometric center of mass with the given weights.

answer:Start by simplifying each part of the expression using the laws of exponents: 1. Simplify (-8^4)^2: [ (-8^4)^2 = (-1)^2 cdot (8^4)^2 = 1 cdot 8^8 = 8^8. ] Thus, (-8^4)^2 = 8^8. 2. Combine the simplified expressions: [frac{1}{(-8^4)^2} cdot (-8)^{11} = frac{1}{8^8} cdot (-8)^{11} = 8^{-8} cdot (-8)^{11}.] 3. Combine exponents of base 8: [ 8^{-8} cdot (-8)^{11} = 8^{-8} cdot (-1)^{11} cdot 8^{11} = -1 cdot 8^{-8+11} = -8^3 = -512. ] Therefore, the simplified expression is [ boxed{-512}. ]

answer:**Answer**: The form of scientific notation is atimes10^{n}, where 1leq|a|<10, and n is an integer. To determine the value of n, we need to see how many places the decimal point has moved to turn the original number into a. The absolute value of n is the same as the number of places the decimal point has moved. When the absolute value of the original number is greater than 1, n is positive; when the absolute value of the original number is less than 1, n is negative. Therefore, to represent 425,000 people in scientific notation, we move the decimal point 5 places to the left, which gives us 4.25times10^{5}. So, the correct answer is boxed{text{D}}.

answer:Substitute x = 4 into the equation: [ f(4) + 3f(-3) = 64. ] -- (1) Next, substitute x = -3: [ f(-3) + 3f(4) = 36. ] -- (2) We can manipulate these equations: Multiply equation (2) by 3: [ 3f(-3) + 9f(4) = 108. ] -- (3) Subtract equation (1) from equation (3): [ 9f(4) - f(4) - 3f(-3) + 3f(-3) = 108 - 64, ] [ 8f(4) = 44, ] [ f(4) = frac{44}{8} = boxed{5.5}. ]