answer：Okay, so I have this complex number z, which is equal to (2 - i) divided by (1 + i). I need to find the real part of z. Hmm, complex numbers can sometimes be tricky, especially when dealing with division. I remember that to simplify expressions like this, you can multiply the numerator and the denominator by the conjugate of the denominator. That should help eliminate the imaginary unit from the denominator.Let me write down the problem again to make sure I have it right: z = (2 - i)/(1 + i). Alright, so the denominator is (1 + i). The conjugate of that would be (1 - i), right? So, if I multiply both the numerator and the denominator by (1 - i), that should help me simplify the expression.Let me do that step by step. First, I'll multiply the numerator (2 - i) by (1 - i). Let me expand that:(2 - i)(1 - i) = 2*1 + 2*(-i) - i*1 - i*(-i) = 2 - 2i - i + i².Wait, i² is equal to -1, so that term becomes -1. So, putting it all together: 2 - 2i - i - 1. Combining like terms, 2 - 1 is 1, and -2i - i is -3i. So, the numerator simplifies to 1 - 3i.Now, let's work on the denominator: (1 + i)(1 - i). That's a difference of squares, so it should be 1² - (i)². Calculating that: 1 - (-1) because i² is -1. So, 1 - (-1) is 1 + 1, which equals 2.So now, after multiplying both numerator and denominator by (1 - i), the expression becomes (1 - 3i)/2. I can separate this into real and imaginary parts: 1/2 - (3/2)i.Therefore, the real part of z is 1/2, and the imaginary part is -3/2. Looking back at the options given, option C is 1/2, so that must be the correct answer.Wait, let me double-check my multiplication to make sure I didn't make a mistake. For the numerator: (2 - i)(1 - i). Let's do it again:First, 2*1 is 2. Then, 2*(-i) is -2i. Next, -i*1 is -i. Finally, -i*(-i) is +i², which is -1. So, adding all those together: 2 - 2i - i - 1. That simplifies to (2 - 1) + (-2i - i) = 1 - 3i. Yeah, that seems right.And the denominator: (1 + i)(1 - i) = 1 - i + i - i². Wait, hold on, I think I did that wrong earlier. Let me recalculate: (1 + i)(1 - i) is 1*(1) + 1*(-i) + i*(1) + i*(-i). So, that's 1 - i + i - i². The -i and +i cancel out, leaving 1 - i². Since i² is -1, that becomes 1 - (-1) = 2. Okay, so that was correct.So, the simplified form is indeed (1 - 3i)/2, which is 1/2 - (3/2)i. So, the real part is 1/2, which is option C. I think that's solid. I don't see any mistakes in my calculations.Just to be thorough, let me plug this back into the original expression to verify. If z = (2 - i)/(1 + i), and we found z = 1/2 - (3/2)i, then multiplying both sides by (1 + i) should give us back (2 - i).So, let's compute (1/2 - 3/2i)(1 + i). Expanding this: 1/2*1 + 1/2*i - 3/2i*1 - 3/2i*i.Calculating each term:1/2*1 = 1/21/2*i = (1/2)i-3/2i*1 = -3/2i-3/2i*i = -3/2i² = -3/2*(-1) = 3/2Now, adding all these together: 1/2 + (1/2)i - 3/2i + 3/2.Combine the real parts: 1/2 + 3/2 = 2.Combine the imaginary parts: (1/2 - 3/2)i = (-1)i.So, altogether, we get 2 - i, which is exactly the numerator we started with. That confirms that our simplified z is correct.Therefore, the real part is indeed 1/2, so the answer is C.**Final Answer**boxed{C}

answer：Okay, so I have this problem where I'm given that ( x + frac{1}{x} = 5 ), and I need to find ( x^3 + frac{1}{x^3} ). Hmm, I remember that when dealing with expressions like this, there are some algebraic identities that can help. Let me think about how to approach this.First, I recall that squaring both sides of the equation might help me find ( x^2 + frac{1}{x^2} ), which could be useful for finding the cube. So, let me try that.Starting with the given equation:[x + frac{1}{x} = 5]If I square both sides, I get:[left( x + frac{1}{x} right)^2 = 5^2]Expanding the left side:[x^2 + 2 cdot x cdot frac{1}{x} + frac{1}{x^2} = 25]Simplifying the middle term:[x^2 + 2 + frac{1}{x^2} = 25]Subtracting 2 from both sides:[x^2 + frac{1}{x^2} = 23]Okay, so now I have ( x^2 + frac{1}{x^2} = 23 ). That's progress. Now, how do I get to ( x^3 + frac{1}{x^3} )?I think there's an identity that relates ( x + frac{1}{x} ) and ( x^2 + frac{1}{x^2} ) to ( x^3 + frac{1}{x^3} ). Let me try to recall it. I believe it's something like:[left( x + frac{1}{x} right) left( x^2 - 1 + frac{1}{x^2} right) = x^3 + frac{1}{x^3}]Wait, is that right? Let me check by expanding the left side:[left( x + frac{1}{x} right) left( x^2 - 1 + frac{1}{x^2} right)]Multiplying term by term:- ( x cdot x^2 = x^3 )- ( x cdot (-1) = -x )- ( x cdot frac{1}{x^2} = frac{1}{x} )- ( frac{1}{x} cdot x^2 = x )- ( frac{1}{x} cdot (-1) = -frac{1}{x} )- ( frac{1}{x} cdot frac{1}{x^2} = frac{1}{x^3} )Now, combining like terms:- ( x^3 ) and ( frac{1}{x^3} ) remain as they are.- ( -x + x = 0 )- ( frac{1}{x} - frac{1}{x} = 0 )So, the entire expression simplifies to:[x^3 + frac{1}{x^3}]Great, so the identity is correct:[left( x + frac{1}{x} right) left( x^2 - 1 + frac{1}{x^2} right) = x^3 + frac{1}{x^3}]Now, I can plug in the known values. From the given, ( x + frac{1}{x} = 5 ), and I found that ( x^2 + frac{1}{x^2} = 23 ). So, substituting into the identity:[5 cdot (23 - 1) = x^3 + frac{1}{x^3}]Wait, hold on. In the identity, it's ( x^2 - 1 + frac{1}{x^2} ), which is ( (x^2 + frac{1}{x^2}) - 1 ). Since ( x^2 + frac{1}{x^2} = 23 ), then:[23 - 1 = 22]So, substituting back:[5 cdot 22 = x^3 + frac{1}{x^3}]Calculating that:[5 times 22 = 110]Therefore, ( x^3 + frac{1}{x^3} = 110 ).Let me just verify my steps to make sure I didn't make any mistakes. Starting from the given equation, squared it correctly to find ( x^2 + frac{1}{x^2} = 23 ). Then, used the identity for the cube, which I expanded and confirmed is correct. Plugged in the known values and did the arithmetic correctly. Seems solid.I think that's it. The answer should be 110.

answer：First, I need to simplify the expression (-2^{-1} times (-8) - sqrt{9} - |-4|).I start by addressing the negative exponent. The term (-2^{-1}) can be rewritten as (-frac{1}{2}).Next, I multiply (-frac{1}{2}) by (-8). Multiplying two negative numbers results in a positive number, so (-frac{1}{2} times (-8) = 4).Then, I evaluate the square root of 9, which is 3.After that, I find the absolute value of -4, which is 4.Now, I substitute these simplified values back into the expression:(4 - 3 - 4).Finally, I perform the subtraction step by step:(4 - 3 = 1),and then (1 - 4 = -3).Therefore, the final answer is (-3).

answer：Okay, so I have this problem here about a cube with edges of length (2sqrt{3}), and all its edges are on the same sphere. I need to find the surface area of that sphere. Hmm, let me try to visualize this. First, I remember that if all the edges of a cube are on a sphere, that means the cube is inscribed in the sphere. So, the sphere touches all the edges of the cube. But wait, actually, when a cube is inscribed in a sphere, all the vertices of the cube lie on the sphere. So, maybe the sphere is circumscribed around the cube, meaning all the cube's vertices are on the sphere. That makes more sense because if all edges are on the sphere, that would mean the sphere is somehow wrapping around the edges, but I think it's more standard to say that the sphere circumscribes the cube, touching all its vertices.So, if the cube is inscribed in the sphere, the sphere's diameter would be equal to the space diagonal of the cube. I remember that the space diagonal of a cube can be calculated using the formula (d = asqrt{3}), where (a) is the edge length of the cube. Let me confirm that. Yeah, for a cube, the space diagonal is indeed (asqrt{3}). Given that the edge length (a) is (2sqrt{3}), let me plug that into the formula. So, the space diagonal (d) would be:[d = asqrt{3} = (2sqrt{3}) times sqrt{3}]Wait, multiplying (sqrt{3}) by (sqrt{3}) gives 3, right? So, that simplifies to:[d = 2 times 3 = 6]So, the space diagonal of the cube is 6 units. Since this diagonal is the diameter of the sphere, the radius (r) of the sphere would be half of that, which is:[r = frac{d}{2} = frac{6}{2} = 3]Alright, now that I have the radius of the sphere, I can find its surface area. The formula for the surface area (S) of a sphere is:[S = 4pi r^2]Substituting (r = 3) into the formula gives:[S = 4pi (3)^2 = 4pi times 9 = 36pi]So, the surface area of the sphere should be (36pi). Let me just double-check my steps to make sure I didn't make any mistakes.1. I identified that the cube is inscribed in the sphere, so the sphere's diameter is the cube's space diagonal.2. Calculated the space diagonal using (d = asqrt{3}), which gave me 6.3. Divided by 2 to get the radius, which is 3.4. Plugged the radius into the surface area formula, resulting in (36pi).Everything seems to check out. I don't think I made any calculation errors. So, the answer should be (36pi), which is option C.

answer：First, I need to understand the formula for calculating the Body Mass Index (BMI). The BMI is defined as the weight in kilograms divided by the square of the height in meters. This means that for any individual, I can calculate their BMI using the formula ( p = frac{w}{h^2} ), where ( w ) is the weight and ( h ) is the height.For Wang Ming, who has a weight of ( w ) kilograms and a height of ( h ) meters, applying the formula directly gives his BMI as ( p = frac{w}{h^2} ).Next, for Zhang Yu, who is 1.80 meters tall and weighs 81 kilograms, I will substitute these values into the BMI formula. First, I calculate the square of the height: ( 1.80^2 = 3.24 ). Then, I divide the weight by this squared height: ( frac{81}{3.24} = 25 ).Finally, I compare this BMI value of 25 to the given categories. Since 25 is greater than 24, Zhang Yu falls into the "Overweight" category.

answer：Okay, so I have this problem about two polynomials. Both are degree 8 polynomials with integer coefficients, and all their roots are integers. The first polynomial is written as ( b_8 x^8 + b_7 x^7 + dots + b_1 x + b_0 = 0 ), and the second one is the reverse of the coefficients: ( b_0 x^8 + b_1 x^7 + dots + b_7 x + b_8 = 0 ). The roots of both polynomials are the same, which are ( s_1, s_2, dots, s_8 ). I need to find the number of possible multisets ( T = {s_1, s_2, dots, s_8} ).Hmm, okay. So both polynomials have the same integer roots, but their coefficients are reversed. I remember that reversing the coefficients of a polynomial relates to the reciprocals of its roots. Let me think about that.If ( p(x) = b_8 x^8 + b_7 x^7 + dots + b_1 x + b_0 ), then the polynomial with reversed coefficients is ( q(x) = b_0 x^8 + b_1 x^7 + dots + b_7 x + b_8 ). I recall that if ( p(x) ) has roots ( s_1, s_2, dots, s_8 ), then ( q(x) ) has roots ( 1/s_1, 1/s_2, dots, 1/s_8 ), provided none of the ( s_i ) are zero. But in this problem, all roots are integers, so ( 1/s_i ) must also be integers because the roots of ( q(x) ) are the same as those of ( p(x) ).Wait, so ( 1/s_i ) must be integers. The only integers whose reciprocals are also integers are 1 and -1. Because if ( s_i ) is any other integer, say 2, then ( 1/2 ) is not an integer. Similarly, for -2, ( 1/(-2) ) is not an integer. So the only possible integer roots are 1 and -1.That simplifies things a lot. So all the roots of both polynomials must be either 1 or -1. Therefore, the multiset ( T ) can only contain 1s and -1s. Now, I need to figure out how many such multisets are possible.Since the polynomial is degree 8, there are 8 roots. Each root is either 1 or -1. So, the multiset is determined by how many 1s and how many -1s there are. Let's say there are ( k ) ones and ( 8 - k ) negative ones, where ( k ) can range from 0 to 8.But wait, I need to make sure that the coefficients of the polynomials are integers. Since all roots are integers, and the coefficients are integers, that's already satisfied because the coefficients are symmetric sums of integers, which are integers themselves.But also, the polynomials have integer coefficients, so when we reverse the coefficients, we must still have the same roots. Since the roots are only 1 and -1, their reciprocals are themselves, so reversing the coefficients doesn't change the roots. So, in this case, the condition is automatically satisfied.Therefore, the number of possible multisets is just the number of ways to choose how many 1s and -1s there are in the multiset. Since ( k ) can be 0, 1, 2, ..., up to 8, that's 9 possibilities.Wait, let me double-check. If ( k = 0 ), all roots are -1. If ( k = 1 ), one root is 1 and seven are -1, and so on, up to ( k = 8 ), all roots are 1. Each of these gives a distinct multiset because the number of 1s and -1s changes. So yes, that's 9 different multisets.I don't think there are any other constraints because the problem only requires that both polynomials have integer coefficients and integer roots, and we've established that the only possible roots are 1 and -1. So, the number of multisets is 9.**Final Answer**The number of possible multisets ( T ) is boxed{9}.