answer：Alright, so I have this expression: √(11 + 6√2) + √(11 - 6√2). The problem says that this value is a natural number, and I need to prove it. Hmm, okay, let me think about how to approach this.First, I remember that sometimes expressions with square roots can be simplified by expressing them as sums or differences of simpler square roots. Maybe I can write each of these square roots as something like √a + √b or √a - √b. Let me try that.Let me denote the first term as A = √(11 + 6√2). Similarly, let me denote the second term as B = √(11 - 6√2). So, the expression we're looking at is A + B.If I can express A and B in terms of simpler square roots, maybe their sum will simplify nicely. Let me assume that A can be written as √x + √y, where x and y are positive real numbers. Similarly, maybe B can be written as √x - √y or something like that. Let me see.So, let's suppose that A = √x + √y. Then, squaring both sides, we get:A² = (√x + √y)² = x + y + 2√(xy)But A² is given as 11 + 6√2. So, we have:x + y + 2√(xy) = 11 + 6√2Similarly, for B, if I suppose that B = √x - √y, then squaring both sides gives:B² = (√x - √y)² = x + y - 2√(xy)And B² is given as 11 - 6√2. So:x + y - 2√(xy) = 11 - 6√2Now, if I add these two equations together:(A² + B²) = [x + y + 2√(xy)] + [x + y - 2√(xy)] = 2x + 2yBut A² + B² is (11 + 6√2) + (11 - 6√2) = 22. So, 2x + 2y = 22, which simplifies to x + y = 11.Similarly, if I subtract the two equations:(A² - B²) = [x + y + 2√(xy)] - [x + y - 2√(xy)] = 4√(xy)But A² - B² is (11 + 6√2) - (11 - 6√2) = 12√2. So, 4√(xy) = 12√2, which simplifies to √(xy) = 3√2, and then squaring both sides gives xy = 9*2 = 18.So now, I have two equations:1. x + y = 112. xy = 18Hmm, so I need to find x and y such that their sum is 11 and their product is 18. This is like solving a quadratic equation. Let me write the quadratic equation whose roots are x and y.Let me denote t² - (x + y)t + xy = 0, which becomes t² - 11t + 18 = 0.Let me solve this quadratic equation. The discriminant D is 121 - 72 = 49. So, the roots are [11 ± √49]/2 = [11 ± 7]/2.So, the roots are (11 + 7)/2 = 18/2 = 9, and (11 - 7)/2 = 4/2 = 2.So, x and y are 9 and 2. Therefore, x = 9 and y = 2, or vice versa.Therefore, going back to A and B:A = √x + √y = √9 + √2 = 3 + √2Similarly, B = √x - √y = √9 - √2 = 3 - √2Wait, hold on. If A = √(11 + 6√2) = 3 + √2, then let me check if that's correct.Let me compute (3 + √2)²: 9 + 6√2 + 2 = 11 + 6√2. Yes, that's correct. Similarly, (3 - √2)² = 9 - 6√2 + 2 = 11 - 6√2. Perfect.So, A = 3 + √2 and B = 3 - √2.Therefore, the expression A + B is (3 + √2) + (3 - √2) = 6. So, the value is 6, which is a natural number. Therefore, we've proven that √(11 + 6√2) + √(11 - 6√2) equals 6, which is indeed a natural number.Wait, let me recap to make sure I didn't skip any steps or make any mistakes. I started by assuming that each square root could be expressed as a sum or difference of simpler square roots. Then, I set up equations based on squaring those expressions and equating them to the given expressions. By adding and subtracting those equations, I was able to find the sum and product of x and y, leading me to solve a quadratic equation. The solutions gave me x and y as 9 and 2, which when plugged back into the expressions for A and B, gave me 3 + √2 and 3 - √2. Adding those together cancels out the √2 terms, leaving me with 6. That seems solid.I think another way to approach this might be to let the entire expression equal some variable and then square both sides. Let me try that method as a check.Let me denote S = √(11 + 6√2) + √(11 - 6√2). Then, S² = [√(11 + 6√2) + √(11 - 6√2)]².Expanding that, we get:S² = (11 + 6√2) + (11 - 6√2) + 2 * √(11 + 6√2) * √(11 - 6√2)Simplify the terms:First, (11 + 6√2) + (11 - 6√2) = 22.Then, the cross term: 2 * √[(11 + 6√2)(11 - 6√2)].Let me compute the product inside the square root:(11 + 6√2)(11 - 6√2) = 11² - (6√2)² = 121 - 36*2 = 121 - 72 = 49.So, the cross term becomes 2 * √49 = 2*7 = 14.Therefore, S² = 22 + 14 = 36.Taking the square root of both sides, S = √36 = 6. Since S is a sum of positive square roots, it must be positive, so S = 6.Yep, that's another way to get to the same result. So, whether I break down each square root individually or consider the entire expression, I end up with the same conclusion that S = 6, which is a natural number.I think both methods are valid and reinforce each other. The first method helps in understanding the structure of each term, while the second method is a more straightforward approach by squaring the entire expression. Either way, the result is consistent.Just to make sure, let me compute the numerical values to verify.Compute √(11 + 6√2):First, compute 6√2: √2 is approximately 1.4142, so 6*1.4142 ≈ 8.4852.So, 11 + 8.4852 ≈ 19.4852. Then, √19.4852 ≈ 4.4142.Similarly, compute √(11 - 6√2):11 - 8.4852 ≈ 2.5148. Then, √2.5148 ≈ 1.5858.Adding these two approximate values: 4.4142 + 1.5858 ≈ 6.0000. So, numerically, it also checks out.Therefore, I'm confident that the expression simplifies to 6, which is a natural number.**Final Answer**The value of the expression is boxed{6}.

answer：First, I need to determine the speed of the students riding bicycles. Let's denote the speed of the bicycles as ( v ) km/h.The bus travels 1.5 times faster than the bicycles, so the speed of the bus is ( 1.5v ) km/h.The distance from the school to the museum is 15 km for both the bicycles and the bus.The students on bicycles leave 15 minutes earlier than the bus. Since 15 minutes is 0.25 hours, the time taken by the bicycles to reach the museum is ( frac{15}{v} ) hours.The time taken by the bus to reach the museum is ( frac{15}{1.5v} ) hours.Since both groups arrive at the same time, the time taken by the bicycles minus the 0.25 hours head start should equal the time taken by the bus. This gives the equation:[frac{15}{v} - 0.25 = frac{15}{1.5v}]Simplifying the equation and solving for ( v ) will give the speed of the bicycles.

answer：First, I need to evaluate each term in the expression separately.Starting with ( (0.25)^{-0.5} ), I recognize that ( 0.25 ) is the same as ( frac{1}{4} ). Applying the negative exponent, this becomes ( 4^{0.5} ). The square root of 4 is 2.Next, for ( left(frac{1}{27}right)^{-frac{1}{3}} ), I rewrite it as ( 27^{frac{1}{3}} ). The cube root of 27 is 3.Finally, evaluating ( 625^{0.25} ), I find the fourth root of 625, which is 5.Now, I substitute these values back into the original expression: ( 2 + 3 - 5 ). Adding 2 and 3 gives 5, and subtracting 5 results in 0.

answer：Okay, so I have this problem here: If (4^{x} = 9^{y} = 6), then find (frac{1}{x} + frac{1}{y}). Hmm, let me try to figure this out step by step.First, I notice that both (4^{x}) and (9^{y}) are equal to 6. That means I can write two separate equations:1. (4^{x} = 6)2. (9^{y} = 6)I need to find (frac{1}{x} + frac{1}{y}). Hmm, maybe I can solve each equation for (x) and (y) separately and then plug them into the expression.Starting with the first equation: (4^{x} = 6). To solve for (x), I can take the logarithm of both sides. I remember that (log(a^b) = b cdot log(a)), so applying that here:(log(4^{x}) = log(6))Which simplifies to:(x cdot log(4) = log(6))Then, solving for (x):(x = frac{log(6)}{log(4)})Okay, got (x). Now, let's do the same for the second equation: (9^{y} = 6).Taking the logarithm of both sides:(log(9^{y}) = log(6))Simplifies to:(y cdot log(9) = log(6))So, solving for (y):(y = frac{log(6)}{log(9)})Alright, now I have expressions for both (x) and (y). The problem asks for (frac{1}{x} + frac{1}{y}). Let me compute each reciprocal first.Starting with (frac{1}{x}):(frac{1}{x} = frac{log(4)}{log(6)})Similarly, (frac{1}{y}):(frac{1}{y} = frac{log(9)}{log(6)})So, adding them together:(frac{1}{x} + frac{1}{y} = frac{log(4)}{log(6)} + frac{log(9)}{log(6)})Since both terms have the same denominator, I can combine them:(frac{log(4) + log(9)}{log(6)})Hmm, I remember that (log(a) + log(b) = log(ab)), so applying that here:(frac{log(4 times 9)}{log(6)})Calculating (4 times 9):(4 times 9 = 36)So, the expression becomes:(frac{log(36)}{log(6)})Wait, (log(36)) can be simplified because 36 is 6 squared. So, (log(36) = log(6^2)). Using the logarithm power rule again:(log(6^2) = 2 cdot log(6))Substituting back into the expression:(frac{2 cdot log(6)}{log(6)})The (log(6)) terms cancel out:(2)So, (frac{1}{x} + frac{1}{y} = 2). Let me double-check my steps to make sure I didn't make any mistakes.1. Started by expressing (x) and (y) in terms of logarithms. That seems correct.2. Took reciprocals of both (x) and (y), which gave me expressions in terms of (log(4)) and (log(9)) over (log(6)).3. Added the reciprocals, combined the logs into a single logarithm of the product, which is 36.4. Recognized that 36 is (6^2), so the logarithm becomes 2 times (log(6)).5. Simplified, and the (log(6)) cancels out, leaving 2.Everything seems to check out. I don't see any errors in my reasoning. So, I think the answer is 2.**Final Answer**boxed{2}

answer：First, let's define the two positive numbers as ( a ) and ( b ), where ( a > b ).According to the problem, the ratio of the two numbers is equal to the ratio of their sum to their difference. This gives us the equation:[frac{a}{b} = frac{a + b}{a - b}]To simplify, we'll cross-multiply:[a(a - b) = b(a + b)]Expanding both sides:[a^2 - ab = ab + b^2]Next, we'll move all terms to one side to set the equation to zero:[a^2 - ab - ab - b^2 = 0]Combining like terms:[a^2 - 2ab - b^2 = 0]This is a quadratic equation in terms of ( a ) and ( b ). To find the ratio ( frac{a}{b} ), we'll divide both sides by ( b^2 ):[left(frac{a}{b}right)^2 - 2left(frac{a}{b}right) - 1 = 0]Let ( x = frac{a}{b} ), then the equation becomes:[x^2 - 2x - 1 = 0]Solving this quadratic equation using the quadratic formula:[x = frac{2 pm sqrt{4 + 4}}{2} = frac{2 pm sqrt{8}}{2} = 1 pm sqrt{2}]Since ( a ) and ( b ) are positive numbers and ( a > b ), the ratio ( frac{a}{b} ) must be positive. Therefore, the valid solution is:[x = 1 + sqrt{2}]

answer：Okay, so Amy and Belinda each have a sheet of paper that's 6 inches by 8 inches. They're both going to roll them up to make cylindrical tubes. Amy is going to tape the two 8-inch sides together, and Belinda is going to tape the two 6-inch sides together. I need to find pi times the positive difference of the volumes of the two tubes. Hmm, okay, let me break this down step by step.First, I remember that when you roll a rectangular sheet into a cylinder, one side becomes the circumference of the base of the cylinder, and the other side becomes the height of the cylinder. So, for Amy, she's taping the two 8-inch sides together. That means the 8-inch side is forming the circumference of the base, right? And the other side, which is 6 inches, becomes the height of her cylinder.Similarly, Belinda is taping the two 6-inch sides together. So, for her, the 6-inch side becomes the circumference of the base, and the 8-inch side becomes the height of her cylinder.Alright, so to find the volume of each cylinder, I know the formula for the volume of a cylinder is V = πr²h, where r is the radius and h is the height. But right now, I have the circumference, not the radius. So, I need to find the radius from the circumference.I remember that the circumference C is related to the radius by the formula C = 2πr. So, if I can find the radius from the circumference, I can plug it into the volume formula.Let me start with Amy's cylinder. Her circumference is 8 inches. So, using C = 2πr, I can solve for r:8 = 2πr Divide both sides by 2π: r = 8 / (2π) Simplify: r = 4 / πOkay, so Amy's radius is 4/π inches. Her height is 6 inches, as given. So, plugging into the volume formula:V_Amy = π * (4/π)² * 6 Let me compute that step by step. First, square the radius: (4/π)² = 16 / π² Multiply by π: π * (16 / π²) = 16 / π Then multiply by the height, which is 6: 16 / π * 6 = 96 / πSo, Amy's volume is 96/π cubic inches.Now, let's do the same for Belinda's cylinder. Her circumference is 6 inches. So, again using C = 2πr:6 = 2πr Divide both sides by 2π: r = 6 / (2π) Simplify: r = 3 / πOkay, Belinda's radius is 3/π inches. Her height is 8 inches. So, plugging into the volume formula:V_Belinda = π * (3/π)² * 8 Again, step by step. Square the radius: (3/π)² = 9 / π² Multiply by π: π * (9 / π²) = 9 / π Multiply by the height, which is 8: 9 / π * 8 = 72 / πSo, Belinda's volume is 72/π cubic inches.Now, the problem asks for pi times the positive difference of the volumes. So, first, let's find the positive difference between the two volumes.V_Amy - V_Belinda = (96/π) - (72/π) = (96 - 72)/π = 24/πSo, the positive difference is 24/π cubic inches. Now, we need to multiply this by π.π * (24/π) = 24Wait, that's interesting. The pi cancels out, leaving just 24. So, pi times the positive difference of the volumes is 24.Let me just double-check my calculations to make sure I didn't make a mistake.For Amy:- Circumference: 8 = 2πr => r = 4/π- Volume: π*(4/π)^2*6 = π*(16/π²)*6 = (16/π)*6 = 96/πFor Belinda:- Circumference: 6 = 2πr => r = 3/π- Volume: π*(3/π)^2*8 = π*(9/π²)*8 = (9/π)*8 = 72/πDifference: 96/π - 72/π = 24/πMultiply by π: 24/π * π = 24Yep, that seems correct. So, the answer is 24.**Final Answer**boxed{24}