This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Step 1: Express m in terms of an integer k. Since m is an odd integer, we can write m = 2k+1 for some integer k. Step 2: Calculate m^2. m^2 = (2k+1)^2 = 4k^2 + 4k + 1 = 4k(k+1) + 1 Since k and k+1 are consecutive integers, one of them must be even. Thus, k(k+1) is always an even number. Let k(k+1) = 2j for some integer j. m^2 = 4(2j) + 1 = 8j + 1 Step 3: Calculate m^4. m^4 = (m^2)^2 = (8j+1)^2 = 64j^2 + 16j + 1 Step 4: Substitute m^2 and m^4 into the given expression. The expression is (m^4 + 4m^2 + 11)/(16). Substitute the values: ((64j^2 + 16j + 1) + 4(8j+1) + 11)/(16) (64j^2 + 16j + 1 + 32j + 4 + 11)/(16) (64j^2 + (16j + 32j) + (1 + 4 + 11))/(16) (64j^2 + 48j + 16)/(16) Step 5: Simplify the expression. (64j^2)/(16) + (48j)/(16) + (16)/(16) 4j^2 + 3j + 1 Since j is an integer, 4j^2 + 3j + 1 is also an integer. Therefore, the number (m^4 + 4m^2 + 11)/(16) is an integer. The final answer is The expression is an integer. Step 1: Combine the terms under a single square root. Given a, b, c are positive real numbers. We use the property sqrt(x) × sqrt(y) = sqrt(xy). sqrt((b)/(a)) × sqrt((c)/(b)) × sqrt((a)/(c)) = sqrt((b)/(a) × (c)/(b) × (a)/(c)) Step 2: Multiply the fractions inside the square root. sqrt((b × c × a)/(a × b × c)) Step 3: Simplify the expression inside the square root. sqrt((abc)/(abc)) = sqrt(1) Step 4: Evaluate the square root. sqrt(1) = 1 Thus, sqrt((b)/(a)) × sqrt((c)/(b)) × sqrt((a)/(c)) = 1. The final answer is 1. Step 1: Substitute the given value of sqrt(n). We are given sqrt(n) = (3)/(2). We need to find -sqrt(n). Step 2: Calculate -sqrt(n). -sqrt(n) = -((3)/(2)) = -(3)/(2) The final answer is -(3)/(2). Step 1: Start with a known true inequality. For any real numbers a and b, the square of their difference is always non-negative: (a-b)^2 0 Step 2: Expand the inequality. a^2 - 2ab + b^2 0 Step 3: Add 2ab to both sides. a^2 + b^2 2ab Step 4: Divide both sides by ab. Since a and b are positive real numbers, ab is also positive. Dividing by a positive number does not change the direction of the inequality. (a^2 + b^2)/(ab) (2ab)/(ab) Step 5: Separate the terms on the left side and simplify. (a^2)/(ab) + (b^2)/(ab) 2 (a)/(b) + (b)/(a) 2 This proves the inequality. The final answer is (a)/(b) + (b)/(a) 2. Step 1: Start with a known true inequality. For any real number x, the square of (x - (1)/(x)) is always non-negative, provided x 0: (x - (1)/(x))^2