💡First, I recognize that the expression (701^2 - 697^2) is a difference of squares. This allows me to use the algebraic identity (a^2 - b^2 = (a + b)(a - b)).Next, I assign (a = 701) and (b = 697). Plugging these values into the identity, I calculate the sum and difference of (a) and (b):[701 + 697 = 1398][701 - 697 = 4]Finally, I multiply these results together to find the value of the original expression:[1398 times 4 = 5592]

💡First, let's determine the unknown number. We know that 30% of 40% of this number equals 36. 1. Calculate 30% of 40%: [ 0.3 times 0.4 = 0.12 ]2. Set up the equation: [ 0.12 times x = 36 ]3. Solve for ( x ): [ x = frac{36}{0.12} = 300 ]Next, find 40% of 30% of the number:1. Calculate 40% of 30%: [ 0.4 times 0.3 = 0.12 ]2. Multiply by the number: [ 0.12 times 300 = 36 ]Finally, calculate 50% of this resultant value:1. Multiply 36 by 50%: [ 0.5 times 36 = 18 ]

💡First, I need to find the side length of the square. Since the perimeter of a square is four times the side length, I can divide the given perimeter by 4 to get the side length in decimeters.Next, I'll calculate the area of the square using the side length I just found. The area of a square is the side length squared.Finally, I need to convert the area from square decimeters to square centimeters. Since 1 square decimeter is equal to 100 square centimeters, I'll multiply the area in square decimeters by 100 to get the area in square centimeters.

💡Okay, so I've got this problem here: The sum of two numbers is 8, and the difference of their squares is 32. I need to find the positive difference between these two numbers. Hmm, let me think about how to approach this.First, I'll call the two numbers x and y. The problem says their sum is 8, so I can write that as:x + y = 8That's straightforward. Now, the second part says the difference of their squares is 32. So, that would be:x² - y² = 32I remember that there's a formula for the difference of squares, which is:x² - y² = (x + y)(x - y)Oh, that's useful! Since I already know that x + y is 8, I can substitute that into the equation:(x + y)(x - y) = 32So, replacing x + y with 8:8(x - y) = 32Now, I can solve for x - y by dividing both sides by 8:x - y = 32 / 8x - y = 4So, the positive difference between the two numbers is 4. That seems straightforward, but let me double-check to make sure I didn't make any mistakes.Let me verify by finding the actual numbers. If x + y = 8 and x - y = 4, I can set up a system of equations:1. x + y = 82. x - y = 4If I add these two equations together:(x + y) + (x - y) = 8 + 42x = 12x = 6Now, substitute x back into the first equation:6 + y = 8y = 2So, the two numbers are 6 and 2. Let's check the difference of their squares:6² - 2² = 36 - 4 = 32Yes, that matches the given condition. So, the positive difference is indeed 4.I think that's correct. I didn't see any errors in my steps, and the verification confirms it. So, the answer should be 4.

💡First, I need to determine the number of nonstudent tickets sold. Since there were a total of 821 tickets sold and 530 of them were student tickets, the number of nonstudent tickets is 821 minus 530, which equals 291.Next, I'll calculate the total revenue generated from student tickets. With 530 student tickets sold at 2 each, the total revenue from students is 530 multiplied by 2, resulting in 1,060.Now, I can find out the total revenue from nonstudent tickets by subtracting the student revenue from the total revenue. The total revenue from nonstudent tickets is 1,933 minus 1,060, which equals 873.Finally, to find the cost of each nonstudent ticket, I'll divide the total nonstudent revenue by the number of nonstudent tickets. Dividing 873 by 291 gives me 3 per nonstudent ticket.

💡Okay, so I need to find the range of the independent variable ( x ) for the function ( y = frac{2}{sqrt{x - 1}} ). Hmm, let me think about this step by step. First, I remember that when dealing with functions, especially ones that have square roots or denominators, there are certain restrictions on the values that ( x ) can take. These restrictions ensure that the function is defined and doesn't involve any mathematical operations that aren't allowed, like dividing by zero or taking the square root of a negative number.So, looking at the function ( y = frac{2}{sqrt{x - 1}} ), I notice two main parts that could impose restrictions on ( x ): the denominator and the square root. Starting with the denominator, since it's in the denominator, the expression ( sqrt{x - 1} ) cannot be zero because division by zero is undefined. That means ( sqrt{x - 1} neq 0 ). If I square both sides of this inequality, I get ( x - 1 neq 0 ), which simplifies to ( x neq 1 ). So, ( x ) can't be equal to 1.Next, considering the square root, the expression inside the square root, which is ( x - 1 ), must be greater than or equal to zero because you can't take the square root of a negative number in the set of real numbers. So, ( x - 1 geq 0 ). Solving this inequality gives ( x geq 1 ).But wait, earlier I found that ( x ) can't be equal to 1 because that would make the denominator zero. So, combining these two results, ( x ) must be greater than 1. Let me double-check that. If ( x = 1 ), then ( sqrt{1 - 1} = sqrt{0} = 0 ), and the function becomes ( y = frac{2}{0} ), which is undefined. If ( x < 1 ), then ( x - 1 ) is negative, and the square root of a negative number isn't a real number, so the function isn't defined there either. Therefore, ( x ) has to be greater than 1 for the function to be defined.Is there anything else I need to consider? Well, the function is ( frac{2}{sqrt{x - 1}} ), so as ( x ) increases beyond 1, ( sqrt{x - 1} ) increases, making the denominator larger and thus ( y ) smaller. But that's about the behavior of the function, not the domain of ( x ).So, putting it all together, the independent variable ( x ) must be greater than 1 for the function to be defined. There are no other restrictions because the numerator is just a constant, 2, which doesn't impose any additional constraints on ( x ).I think that's it. The range of ( x ) is all real numbers greater than 1.