answer：First, I need to calculate the expression inside the parentheses: (18 times 23 - 24 times 17).I'll start by multiplying 18 by 23, which equals 414.Next, I'll multiply 24 by 17, resulting in 408.Now, I'll subtract 408 from 414 to get 6.Then, I'll divide this result by 3, which gives me 2.Finally, I'll add 5 to 2, resulting in the final answer of 7.

answer：Okay, so I have this problem about an even function f(x) that's defined on all real numbers. It says that for x greater than or equal to 0, f(x + 2) equals f(x), which means it's periodic with period 2 on that interval. And when x is in [0, 2), f(x) is given as log base 2 of (x + 1). I need to find f(-2010) + f(2011). Hmm, let me try to break this down step by step.First, since f(x) is an even function, I remember that means f(-x) = f(x) for all x. So, f(-2010) is the same as f(2010). That simplifies things a bit because now I just need to find f(2010) and f(2011) and add them together.Next, the function is periodic with period 2 for x >= 0. So, f(x + 2) = f(x) for x >= 0. That means the function repeats every 2 units on the positive side. So, to find f(2010) and f(2011), I can use this periodicity to reduce the arguments to within the interval [0, 2), where the function is defined explicitly.Let me start with f(2010). Since 2010 is a positive number, I can use the periodicity. The period is 2, so I can divide 2010 by 2 to find how many periods fit into 2010. 2010 divided by 2 is 1005, which is an integer. So, 2010 is exactly 1005 periods away from 0. Therefore, f(2010) = f(0). Now, looking at the definition of f(x) on [0, 2), f(0) is log base 2 of (0 + 1) which is log base 2 of 1. I remember that log base 2 of 1 is 0 because 2^0 = 1. So, f(0) = 0. Therefore, f(2010) = 0.Now, moving on to f(2011). Again, 2011 is positive, so I can use the periodicity. Let me divide 2011 by 2 to find the remainder. 2011 divided by 2 is 1005 with a remainder of 1. So, 2011 = 2*1005 + 1. Therefore, f(2011) = f(1). Looking at the definition on [0, 2), f(1) is log base 2 of (1 + 1) which is log base 2 of 2. I know that log base 2 of 2 is 1 because 2^1 = 2. So, f(1) = 1. Therefore, f(2011) = 1.So, putting it all together, f(-2010) is equal to f(2010), which is 0, and f(2011) is 1. Therefore, f(-2010) + f(2011) is 0 + 1, which equals 1.Wait, let me double-check my reasoning to make sure I didn't make any mistakes. First, f(x) is even, so f(-x) = f(x). That part seems straightforward. Then, for x >= 0, f is periodic with period 2. So, f(x + 2) = f(x). That means every time I add 2 to x, the function repeats its values. So, for any integer multiple of 2 added to x, the function value remains the same.For f(2010), since 2010 is divisible by 2, it's exactly 1005 periods away from 0. So, f(2010) = f(0). And f(0) is log base 2 of 1, which is 0. That seems correct.For f(2011), 2011 divided by 2 is 1005.5, but since we're dealing with integer periods, the remainder is 1. So, f(2011) = f(1). f(1) is log base 2 of 2, which is 1. That also seems correct.Therefore, adding f(-2010) and f(2011) gives 0 + 1 = 1. I think that's the right answer.Just to make sure, let me consider another approach. Maybe using modular arithmetic to find the equivalent x in [0, 2). For f(2010), 2010 mod 2 is 0, so f(2010) = f(0) = 0. For f(2011), 2011 mod 2 is 1, so f(2011) = f(1) = 1. Yep, same result.Alternatively, I can think about the graph of the function. Since it's even, it's symmetric about the y-axis. For x >= 0, it's periodic with period 2, so every 2 units, it repeats. So, the function from 0 to 2 is log base 2 of (x + 1), which is a logarithmic curve starting at (0, 0) and going up to (2, log2(3)). Then, it repeats every 2 units. So, at x = 2010, which is a multiple of 2, it's at the same point as x = 0, which is 0. At x = 2011, it's at the same point as x = 1, which is 1. So, the sum is 1.I think I'm confident that the answer is 1.**Final Answer**boxed{1}

answer：First, I need to determine the initial number of socks in Jamal's drawer. There are 6 green socks, 18 purple socks, and 12 orange socks. Adding these together gives a total of 36 socks.Let’s denote the number of purple socks Jamal adds as ( x ). After adding these socks, the total number of socks becomes ( 36 + x ), and the number of purple socks becomes ( 18 + x ).According to the problem, the probability of picking a purple sock is 60%, or 0.6. This can be expressed as the ratio of purple socks to the total number of socks:[frac{18 + x}{36 + x} = 0.6]To solve for ( x ), I'll multiply both sides of the equation by ( 36 + x ):[18 + x = 0.6(36 + x)]Expanding the right side:[18 + x = 21.6 + 0.6x]Subtracting ( 0.6x ) from both sides:[18 + 0.4x = 21.6]Subtracting 18 from both sides:[0.4x = 3.6]Finally, dividing both sides by 0.4:[x = 9]Therefore, Jamal added a total of 9 purple socks.

answer：To determine the slope of the tangent to the curve ( y = e^x ) at the origin, I first need to find the derivative of the function, which represents the slope of the tangent at any point ( x ).The derivative of ( e^x ) with respect to ( x ) is ( e^x ). Therefore, the slope of the tangent at any point ( x ) is ( e^x ).Next, I evaluate this derivative at the origin, where ( x = 0 ). Substituting ( x = 0 ) into the derivative gives ( e^0 = 1 ).Thus, the slope of the tangent to the curve ( y = e^x ) at the origin is 1.

answer：First, I need to calculate the difference between ( a ) and ( b ). Given ( a = 0.5 ) and ( b = frac{1}{3} ), I'll subtract ( b ) from ( a ).Next, I'll convert ( 0.5 ) to a fraction to make the subtraction easier. ( 0.5 ) is equivalent to ( frac{1}{2} ).Now, I'll find a common denominator for ( frac{1}{2} ) and ( frac{1}{3} ). The least common denominator is 6. Converting both fractions, ( frac{1}{2} ) becomes ( frac{3}{6} ) and ( frac{1}{3} ) becomes ( frac{2}{6} ).Subtracting these gives ( frac{3}{6} - frac{2}{6} = frac{1}{6} ).Finally, I'll determine how many times ( frac{1}{78} ) fits into ( frac{1}{6} ). Dividing ( frac{1}{6} ) by ( frac{1}{78} ) results in ( 13 ).

answer：To find the zero of the function ( y = lg x - 1 ), I need to determine the value of ( x ) that makes ( y = 0 ).First, I'll set the function equal to zero:[lg x - 1 = 0]Next, I'll isolate ( lg x ) by adding 1 to both sides of the equation:[lg x = 1]Since ( lg x ) represents the logarithm base 10 of ( x ), I can rewrite the equation in exponential form:[x = 10^1]Finally, calculating ( 10^1 ) gives:[x = 10]