answer：First, I need to find the sum of the first 25 positive even integers. The formula for the sum of the first ( n ) even integers is ( n(n + 1) ). Plugging in ( n = 25 ), the sum is ( 25 times 26 = 650 ).Next, I need to express this sum as the sum of five consecutive even integers. Let the smallest of these integers be ( x ). The five consecutive even integers can then be written as ( x ), ( x + 2 ), ( x + 4 ), ( x + 6 ), and ( x + 8 ).The sum of these five integers is:[x + (x + 2) + (x + 4) + (x + 6) + (x + 8) = 5x + 20]Setting this equal to 650:[5x + 20 = 650]Solving for ( x ):[5x = 630 x = 126]The five consecutive even integers are 126, 128, 130, 132, and 134. The largest of these integers is 134.

answer：First, determine the cost of one dozen apples by dividing the total cost by the number of dozens:[text{Cost per dozen} = frac{15.60}{2} = 7.80]Next, calculate the cost for four dozen apples by multiplying the cost per dozen by four:[text{Cost for four dozen} = 4 times 7.80 = 31.20]Finally, present the calculated cost in a boxed format:[boxed{31.20}]

answer：First, I need to add the numbers 3456, 4563, 5634, and 6345. To make this easier, I'll align them vertically and add each column starting from the rightmost digit.Starting with the units column: 6 + 3 + 4 + 5 equals 18. I'll write down 8 and carry over 1 to the next column.Next, in the tens column: 5 + 6 + 3 + 4 equals 18, plus the carried over 1 makes 19. I'll write down 9 and carry over 1.Moving to the hundreds column: 4 + 5 + 6 + 3 equals 18, plus the carried over 1 makes 19. I'll write down 9 and carry over 1.Finally, in the thousands column: 3 + 4 + 5 + 6 equals 18, plus the carried over 1 makes 19. I'll write down 19.Putting it all together, the sum of these numbers is 19998.

answer：To evaluate ( log_{16}64 ), I start by letting ( x = log_{16}64 ). According to the definition of logarithms, this means that ( 16^x = 64 ).Next, I express both 16 and 64 as powers of 2, which gives me ( 16 = 2^4 ) and ( 64 = 2^6 ). Substituting these into the equation, I have ( (2^4)^x = 2^6 ).Simplifying the left side using the power of a power property, I get ( 2^{4x} = 2^6 ). Since the bases are the same, I can set the exponents equal to each other, resulting in ( 4x = 6 ).Solving for ( x ), I divide both sides by 4, which gives ( x = frac{6}{4} = frac{3}{2} ).Therefore, ( log_{16}64 = frac{3}{2} ).

answer：First, I need to factor the given polynomial ( 45x^6 - 270x^{12} + 90x^7 ).I'll start by identifying the greatest common factor (GCF) of the coefficients and the variables.For the coefficients:- ( 45 = 5 times 9 = 5 times 3^2 )- ( 270 = 5 times 54 = 5 times 3 times 18 = 5 times 3^3 times 2 )- ( 90 = 5 times 18 = 5 times 3^2 times 2 )The common factors among the coefficients are ( 5 times 3^2 = 45 ).For the variables, the smallest power of ( x ) present in all terms is ( x^6 ).So, the GCF is ( 45x^6 ).Next, I'll factor out ( 45x^6 ) from each term in the polynomial:[45x^6 - 270x^{12} + 90x^7 = 45x^6(1 - 6x^6 + 2x)]Finally, I'll rearrange the terms inside the parentheses for clarity:[45x^6(1 + 2x - 6x^6)]

answer：First, I recognize that the area of a circle is given by the formula ( pi r^2 ), where ( r ) is the radius.Given the area is ( 16pi ), I can set up the equation ( pi r^2 = 16pi ).By dividing both sides of the equation by ( pi ), I simplify it to ( r^2 = 16 ).Taking the square root of both sides gives me ( r = 4 ).The diameter ( d ) of the circle is twice the radius, so ( d = 2 times 4 = 8 ).Finally, to find three times the diameter, I calculate ( 3 times 8 = 24 ).