💡First, I need to find the expressions for **_a_** + **_b_** and 2**_a_** - **_b_**.Given **_a_** = (2, 1) and **_b_** = (_x_, -2), I can calculate:**_a_** + **_b_** = (2 + _x_, 1 + (-2)) = (2 + _x_, -1)2**_a_** - **_b_** = (2*2 - _x_, 2*1 - (-2)) = (4 - _x_, 4)Since **_a_** + **_b_** is parallel to 2**_a_** - **_b_**, their cross product must be zero:(2 + _x_)(4) - (-1)(4 - _x_) = 0Expanding this equation:8 + 4_x_ + 4 - _x_ = 0Combining like terms:12 + 3_x_ = 0Solving for _x_:3_x_ = -12_x_ = -4Therefore, the value of _x_ is -4.

💡Okay, so I have this cube, right? It's labeled ABCD_A1B1C1D1. Hmm, that's a bit confusing with all those subscripts, but I think I can figure it out. The problem says that the edges AB, AD, and AA1 lie on the axes of a three-dimensional Cartesian coordinate system. So, that means AB is along the x-axis, AD is along the y-axis, and AA1 is along the z-axis. Got it.The cube has an edge length of one unit. So, each edge—AB, AD, AA1, etc.—is one unit long. I need to find the coordinates of the midpoint of edge CC1. Alright, let's break this down step by step.First, I should probably figure out the coordinates of all the relevant points in this cube. Since AB, AD, and AA1 are along the x, y, and z axes respectively, and the cube has edge length one, I can assign coordinates to the vertices based on that.Let's start by assigning coordinates to point A. Since AB is along the x-axis, AD along the y-axis, and AA1 along the z-axis, point A is at the origin. So, A is at (0, 0, 0).Now, since AB is along the x-axis and has length one, point B must be at (1, 0, 0). Similarly, AD is along the y-axis, so point D is at (0, 1, 0). And AA1 is along the z-axis, so point A1 is at (0, 0, 1).Next, let's figure out the coordinates of point C. In a cube, point C is diagonally opposite to point A on the base face ABCD. Since AB is along the x-axis and AD is along the y-axis, moving one unit in both the x and y directions from point A should give us point C. So, point C is at (1, 1, 0).Similarly, point C1 is the top counterpart of point C in the cube. Since AA1 is along the z-axis, moving one unit up from point C along the z-axis should give us point C1. So, point C1 is at (1, 1, 1).Now, I need to find the midpoint of edge CC1. The midpoint of a line segment in three-dimensional space can be found by taking the average of the corresponding coordinates of the endpoints. So, if point C is at (1, 1, 0) and point C1 is at (1, 1, 1), the midpoint M will have coordinates:M_x = (1 + 1)/2 = 1M_y = (1 + 1)/2 = 1M_z = (0 + 1)/2 = 0.5So, the coordinates of the midpoint M are (1, 1, 0.5).Wait, let me double-check that. If I move from point C (1,1,0) halfway towards point C1 (1,1,1), I should indeed end up at (1,1,0.5). Yeah, that makes sense because halfway between 0 and 1 on the z-axis is 0.5.Is there anything else I need to consider here? Maybe the orientation of the cube or the labeling of the vertices? Let me think. The cube is labeled ABCD_A1B1C1D1, which suggests that each vertex has a corresponding top vertex with a subscript 1. So, A corresponds to A1, B to B1, and so on. That aligns with how I assigned the coordinates earlier.Just to visualize, if I imagine the cube with A at the origin, then moving along the x-axis gets me to B, along the y-axis to D, and up the z-axis to A1. From there, the other points like C and C1 are just combinations of these movements. So, point C is at (1,1,0), and C1 is directly above it at (1,1,1).Calculating the midpoint again: (1+1)/2=1, (1+1)/2=1, (0+1)/2=0.5. Yep, that seems consistent. I don't think I made any mistakes here.I guess another way to think about it is that since CC1 is a vertical edge (along the z-axis) of the cube, its midpoint should be halfway up that edge. Since the cube has a height of one unit, halfway would be at 0.5 units above the base. So, in terms of coordinates, that translates to z=0.5, while the x and y coordinates remain the same as point C, which are both 1.So, putting it all together, the midpoint of edge CC1 is at (1,1,0.5). I think that's the correct answer.**Final Answer**The coordinates of the midpoint of edge ( CC_1 ) are boxed{left(1, 1, frac{1}{2}right)}.

💡Okay, so I've got these two systems of linear equations to solve. Let me take them one at a time and think through how to approach each.Starting with the first system:[left{begin{array}{l}y = x + 1 x + y = 5end{array}right.]Hmm, the first equation is already solved for y, which might make substitution a good method here. If I plug the expression for y from the first equation into the second equation, I can solve for x first and then find y. Let's try that.So, substituting y = x + 1 into the second equation:[x + (x + 1) = 5]Simplifying that:[2x + 1 = 5]Subtracting 1 from both sides:[2x = 4]Dividing both sides by 2:[x = 2]Now that I have x, I can plug it back into the first equation to find y:[y = 2 + 1 = 3]Alright, so for the first system, x is 2 and y is 3. That seems straightforward.Moving on to the second system:[left{begin{array}{l}x + 2y = 9 3x - 2y = -1end{array}right.]Looking at these equations, I notice that the coefficients of y are opposites (2 and -2). This might be a good candidate for the elimination method. If I add the two equations together, the y terms should cancel out.Adding the equations:[(x + 2y) + (3x - 2y) = 9 + (-1)]Simplifying:[4x = 8]Dividing both sides by 4:[x = 2]Now that I have x, I can substitute it back into one of the original equations to find y. Let's use the first equation:[2 + 2y = 9]Subtracting 2 from both sides:[2y = 7]Dividing both sides by 2:[y = 3.5]So, for the second system, x is 2 and y is 3.5.Wait a second, both systems have x = 2? That's interesting. Let me double-check my work to make sure I didn't make a mistake.For the first system:- Substituted y = x + 1 into x + y = 5 correctly.- Calculated x = 2, then y = 3. Seems right.For the second system:- Added the two equations to eliminate y, got x = 2.- Plugged x = 2 back into the first equation, solved for y = 3.5. That also seems correct.Okay, so both systems happen to have the same x-value, but different y-values. That makes sense because they're different systems. I think my solutions are correct.

💡First, recognize that the minute hand completes a full circle, which is (2pi) radians, in 60 minutes.Next, determine the angle covered by the minute hand in one minute by dividing the total angle by the total time:[frac{2pi text{ radians}}{60 text{ minutes}} = frac{pi}{30} text{ radians per minute}]Then, calculate the angle turned when the minute hand is moved back by 10 minutes by multiplying the angle per minute by 10:[10 text{ minutes} times frac{pi}{30} text{ radians per minute} = frac{pi}{3} text{ radians}]Finally, conclude that the minute hand has turned an angle of (frac{pi}{3}) radians.

💡Okay, so I have this problem about a geometric sequence. The first term is 512, and the eighth term is 2. I need to find the positive, real value for the sixth term. Hmm, let me think about how to approach this.First, I remember that in a geometric sequence, each term is found by multiplying the previous term by a common ratio, which we usually denote as 'r'. So, the nth term of a geometric sequence can be expressed as:[ a_n = a_1 times r^{n-1} ]Where:- ( a_n ) is the nth term,- ( a_1 ) is the first term,- ( r ) is the common ratio,- ( n ) is the term number.Given that the first term ( a_1 ) is 512, and the eighth term ( a_8 ) is 2, I can set up an equation using the formula for the nth term.So, for the eighth term:[ a_8 = 512 times r^{8-1} ][ 2 = 512 times r^7 ]Now, I need to solve for 'r'. Let me rearrange the equation:[ r^7 = frac{2}{512} ]Simplifying the fraction on the right side:[ frac{2}{512} = frac{1}{256} ]So,[ r^7 = frac{1}{256} ]To find 'r', I need to take the seventh root of both sides. That is,[ r = sqrt[7]{frac{1}{256}} ]Hmm, I know that 256 is a power of 2. Let me confirm:[ 2^8 = 256 ]Yes, that's correct. So,[ frac{1}{256} = 2^{-8} ]Therefore,[ r = sqrt[7]{2^{-8}} ]Using the property of exponents that ( sqrt[n]{a^m} = a^{m/n} ), this becomes:[ r = 2^{-8/7} ]Which can also be written as:[ r = left(2^{-1}right)^{8/7} = left(frac{1}{2}right)^{8/7} ]But since we're looking for the positive, real value, and knowing that 2 is a positive real number, this is acceptable.Now, I need to find the sixth term ( a_6 ). Using the nth term formula again:[ a_6 = 512 times r^{6-1} = 512 times r^5 ]Substituting the value of 'r' we found:[ a_6 = 512 times left(2^{-8/7}right)^5 ]Simplify the exponent:[ left(2^{-8/7}right)^5 = 2^{-40/7} ]So,[ a_6 = 512 times 2^{-40/7} ]Hmm, 512 is also a power of 2. Let me express 512 as a power of 2:[ 512 = 2^9 ]So, substituting back:[ a_6 = 2^9 times 2^{-40/7} ]Using the property of exponents that ( a^m times a^n = a^{m+n} ):[ a_6 = 2^{9 - 40/7} ]Let me compute the exponent:First, convert 9 to sevenths:[ 9 = frac{63}{7} ]So,[ 9 - frac{40}{7} = frac{63}{7} - frac{40}{7} = frac{23}{7} ]Therefore,[ a_6 = 2^{23/7} ]Hmm, 23/7 is approximately 3.2857, but I need an exact value. Let me see if I can express this differently.Alternatively, perhaps I made a miscalculation earlier. Let me double-check my steps.Starting from:[ r^7 = frac{1}{256} ][ r = left(frac{1}{256}right)^{1/7} ][ r = left(2^{-8}right)^{1/7} ][ r = 2^{-8/7} ]That seems correct.Then for ( a_6 ):[ a_6 = 512 times r^5 ][ = 2^9 times left(2^{-8/7}right)^5 ][ = 2^9 times 2^{-40/7} ][ = 2^{9 - 40/7} ][ = 2^{(63/7 - 40/7)} ][ = 2^{23/7} ]Yes, that's correct. So, ( a_6 = 2^{23/7} ). But the problem asks for the positive, real value. I can express this as:[ 2^{23/7} = sqrt[7]{2^{23}} ]But 23 divided by 7 is 3 with a remainder of 2, so:[ 2^{23/7} = 2^{3 + 2/7} = 2^3 times 2^{2/7} = 8 times sqrt[7]{2^2} = 8 times sqrt[7]{4} ]Hmm, that might not be the simplest form. Alternatively, perhaps I can find a numerical approximation, but since the problem asks for the exact value, I should leave it in exponential form.Wait, maybe I made a mistake earlier in calculating the exponent. Let me check:Starting from ( a_6 = 512 times r^5 ), and ( r = 2^{-8/7} ).So,[ a_6 = 512 times left(2^{-8/7}right)^5 ][ = 512 times 2^{-40/7} ][ = 2^9 times 2^{-40/7} ][ = 2^{9 - 40/7} ][ = 2^{(63/7 - 40/7)} ][ = 2^{23/7} ]Yes, that's correct. So, ( a_6 = 2^{23/7} ). But let me see if this can be simplified further or expressed differently.Alternatively, perhaps I can express 23/7 as a mixed number:[ 23/7 = 3 frac{2}{7} ]So,[ 2^{23/7} = 2^{3 + 2/7} = 2^3 times 2^{2/7} = 8 times 2^{2/7} ]But ( 2^{2/7} ) is the seventh root of ( 2^2 ), which is the seventh root of 4. So,[ 2^{23/7} = 8 times sqrt[7]{4} ]I think that's as simplified as it gets. However, the problem asks for the positive, real value, so this is acceptable.Wait, but earlier in the assistant's solution, they found ( r = 1/2 ) and then calculated ( a_6 = 16 ). But according to my calculations, ( a_6 = 2^{23/7} ), which is approximately 16. Let me check:Calculating ( 2^{23/7} ):First, 23 divided by 7 is approximately 3.2857.So, ( 2^{3.2857} ) is approximately:We know that ( 2^3 = 8 ), ( 2^4 = 16 ).Since 3.2857 is closer to 3 than to 4, but let's compute it more accurately.Using logarithms or a calculator:( ln(2^{23/7}) = (23/7) ln 2 approx (3.2857)(0.6931) approx 2.281 )So, ( e^{2.281} approx 9.8 ). Wait, that can't be right because ( 2^3 = 8 ) and ( 2^{3.2857} ) should be higher than 8 but less than 16.Wait, no, my mistake. Actually, ( 2^{23/7} ) is approximately:Let me compute 23/7 ≈ 3.2857So, 2^3 = 82^0.2857 ≈ ?We know that 2^(1/7) ≈ 1.104, so 2^(2/7) ≈ (1.104)^2 ≈ 1.219Therefore, 2^3.2857 = 2^3 * 2^0.2857 ≈ 8 * 1.219 ≈ 9.752Wait, that's about 9.75, but the assistant's solution got 16. That's a discrepancy. So, where did I go wrong?Wait, let's go back. The assistant found r = 1/2, which would make sense because (1/2)^7 = 1/128, but wait, 512*(1/2)^7 = 512*(1/128) = 4, not 2. Wait, that's not matching.Wait, no, let's see:If r = 1/2, then r^7 = (1/2)^7 = 1/128So, 512 * (1/128) = 4, but the eighth term is supposed to be 2. So, that's incorrect.Wait, so the assistant's solution must be wrong because they got r = 1/2, but that leads to a_8 = 4, not 2.So, that means my initial approach was correct, and the assistant made a mistake.Wait, let me check the assistant's solution again.They said:r^7 = 1/256Then r = (1/256)^(1/7) = 1/2But 256 is 2^8, so (2^8)^(1/7) = 2^(8/7), so 1/256 is 2^-8, so (2^-8)^(1/7) = 2^(-8/7), which is approximately 0.5, but not exactly 1/2.Wait, 2^(8/7) is approximately 2^(1.1428) ≈ 2.24, so 1/2.24 ≈ 0.446, not 0.5.So, the assistant incorrectly took the seventh root of 1/256 as 1/2, but actually, it's approximately 0.446.Therefore, the correct value of r is 2^(-8/7), which is approximately 0.446.Then, proceeding to find a_6:a_6 = 512 * r^5 = 512 * (2^(-8/7))^5 = 512 * 2^(-40/7)Since 512 = 2^9, this becomes:2^9 * 2^(-40/7) = 2^(9 - 40/7) = 2^(63/7 - 40/7) = 2^(23/7)Which is approximately 2^3.2857 ≈ 9.75But the assistant got 16, which is incorrect because their value of r was wrong.So, the correct answer should be 2^(23/7), which is approximately 9.75, but since the problem asks for the exact value, it's 2^(23/7).Alternatively, perhaps I can express this as 8 * 2^(2/7), as I did earlier.But let me see if there's another way to approach this problem to confirm.Alternatively, since the eighth term is 2, and the first term is 512, the ratio r can be found by:r^7 = 2 / 512 = 1/256So, r = (1/256)^(1/7)But 256 = 2^8, so:r = (2^8)^(1/7) = 2^(8/7)Wait, no, that's r^7 = 1/256 = 2^-8So, r = (2^-8)^(1/7) = 2^(-8/7)Yes, that's correct.So, r = 2^(-8/7)Then, a_6 = 512 * r^5 = 2^9 * (2^(-8/7))^5 = 2^9 * 2^(-40/7) = 2^(63/7 - 40/7) = 2^(23/7)So, yes, that's correct.Therefore, the sixth term is 2^(23/7), which is approximately 9.75, but the exact value is 2^(23/7).Wait, but the problem asks for the positive, real value, so 2^(23/7) is positive and real, so that's acceptable.Alternatively, perhaps I can express this as 8 * 2^(2/7), as I did earlier.But I think 2^(23/7) is the simplest exact form.Wait, but let me check if 23/7 can be simplified or if there's a better way to express this.23 and 7 are both primes, so 23/7 is already in simplest terms.Therefore, the exact value is 2^(23/7).But let me see if the problem expects a different form or if I made a mistake in the calculation.Wait, another approach: perhaps using logarithms to solve for r.Given:512 * r^7 = 2So,r^7 = 2 / 512 = 1/256Taking natural logarithm on both sides:7 ln r = ln(1/256) = -ln(256) = -8 ln 2Therefore,ln r = (-8 ln 2)/7So,r = e^(-8 ln 2 /7) = (e^{ln 2})^{-8/7} = 2^{-8/7}Which is the same as before.Therefore, r = 2^{-8/7}Then, a_6 = 512 * r^5 = 2^9 * (2^{-8/7})^5 = 2^9 * 2^{-40/7} = 2^{63/7 - 40/7} = 2^{23/7}So, yes, that's correct.Therefore, the sixth term is 2^{23/7}.But wait, 23/7 is approximately 3.2857, so 2^3.2857 is approximately 9.75, but the problem might expect an exact value, so 2^{23/7} is the exact value.Alternatively, perhaps I can express this as 8 * 2^{2/7}, since 23/7 = 3 + 2/7, so 2^{3 + 2/7} = 2^3 * 2^{2/7} = 8 * 2^{2/7}But 2^{2/7} is the seventh root of 4, so:a_6 = 8 * sqrt[7]{4}Which is another way to write it.But I think 2^{23/7} is acceptable.Alternatively, perhaps the problem expects a different approach or a different form.Wait, another thought: maybe the problem expects the answer in terms of exponents with base 2, but simplified.Given that 23/7 is approximately 3.2857, but perhaps there's a way to express it as a combination of exponents.But I don't think so. 23 is a prime number, so it can't be broken down further in terms of exponents.Therefore, I think the exact value is 2^{23/7}, which is approximately 9.75.But let me check the assistant's solution again, because they got 16, which is 2^4.Wait, perhaps they made a mistake in calculating r.They said:r^7 = 1/256Then, r = (1/256)^{1/7} = 1/2But 256 is 2^8, so (2^8)^{1/7} = 2^{8/7}, so 1/256 is 2^{-8}, so (2^{-8})^{1/7} = 2^{-8/7}Therefore, r = 2^{-8/7}, not 1/2.So, the assistant incorrectly took the seventh root of 1/256 as 1/2, but actually, it's 2^{-8/7}.Therefore, their value of r was wrong, leading to an incorrect a_6.So, the correct value of r is 2^{-8/7}, and thus a_6 is 2^{23/7}, which is approximately 9.75.Therefore, the answer should be 2^{23/7}, or 8 * 2^{2/7}, or 8 * sqrt[7]{4}.But perhaps the problem expects a numerical approximation, but since it's a math problem, the exact value is preferred.Therefore, the positive, real value for the sixth term is 2^{23/7}.Alternatively, if I rationalize it differently, perhaps expressing it as 8 * 2^{2/7}, but both are equivalent.So, I think that's the correct answer.

💡To determine the number of dots on level ( Z ), we start with level ( A ) which has 1 dot. The pattern alternates between doubling the number of dots and keeping the same number of dots as the previous level. 1. **Level ( A )**: 1 dot2. **Level ( B )**: ( 2 times 1 = 2 ) dots3. **Level ( C )**: Same as level ( B ), so 2 dots4. **Level ( D )**: ( 2 times 2 = 4 ) dots5. **Level ( E )**: Same as level ( D ), so 4 dots6. **Level ( F )**: ( 2 times 4 = 8 ) dots7. **Level ( G )**: Same as level ( F ), so 8 dots8. **Level ( H )**: ( 2 times 8 = 16 ) dots9. **Level ( I )**: Same as level ( H ), so 16 dots10. **Level ( J )**: ( 2 times 16 = 32 ) dots11. **Level ( K )**: Same as level ( J ), so 32 dots12. **Level ( L )**: ( 2 times 32 = 64 ) dots13. **Level ( M )**: Same as level ( L ), so 64 dots14. **Level ( N )**: ( 2 times 64 = 128 ) dots15. **Level ( O )**: Same as level ( N ), so 128 dots16. **Level ( P )**: ( 2 times 128 = 256 ) dots17. **Level ( Q )**: Same as level ( P ), so 256 dots18. **Level ( R )**: ( 2 times 256 = 512 ) dots19. **Level ( S )**: Same as level ( R ), so 512 dots20. **Level ( T )**: ( 2 times 512 = 1024 ) dots21. **Level ( U )**: Same as level ( T ), so 1024 dots22. **Level ( V )**: ( 2 times 1024 = 2048 ) dots23. **Level ( W )**: Same as level ( V ), so 2048 dots24. **Level ( X )**: ( 2 times 2048 = 4096 ) dots25. **Level ( Y )**: Same as level ( X ), so 4096 dots26. **Level ( Z )**: ( 2 times 4096 = 8192 ) dotsThus, the number of dots on level ( Z ) is (boxed{8192}).