💡Okay, so I have this problem here: I need to find the value of ( k ) in the expression [ (5^{1002} + 6^{1001})^2 - (5^{1002} - 6^{1001})^2 = k cdot 30^{1001}. ] Hmm, that looks a bit intimidating with all those exponents, but maybe I can simplify it somehow.First, I remember that there's a formula for the difference of squares: ( a^2 - b^2 = (a - b)(a + b) ). Maybe I can apply that here. Let me set ( a = 5^{1002} + 6^{1001} ) and ( b = 5^{1002} - 6^{1001} ). Then the expression becomes ( a^2 - b^2 ), which factors into ( (a - b)(a + b) ).Let me compute ( a - b ) and ( a + b ). Starting with ( a - b ):[ (5^{1002} + 6^{1001}) - (5^{1002} - 6^{1001}) = 5^{1002} + 6^{1001} - 5^{1002} + 6^{1001} = 2 cdot 6^{1001} ]Now, ( a + b ):[ (5^{1002} + 6^{1001}) + (5^{1002} - 6^{1001}) = 5^{1002} + 6^{1001} + 5^{1002} - 6^{1001} = 2 cdot 5^{1002} ]So, putting it all together:[ (a - b)(a + b) = (2 cdot 6^{1001})(2 cdot 5^{1002}) = 4 cdot 5^{1002} cdot 6^{1001} ]Alright, so the left side of the equation simplifies to ( 4 cdot 5^{1002} cdot 6^{1001} ). Now, I need to express this as ( k cdot 30^{1001} ).Let me see. I know that ( 30 = 5 cdot 6 ), so ( 30^{1001} = (5 cdot 6)^{1001} = 5^{1001} cdot 6^{1001} ).Looking back at my expression, I have ( 4 cdot 5^{1002} cdot 6^{1001} ). Let me break down the exponents:- ( 5^{1002} = 5^{1001} cdot 5 )- ( 6^{1001} ) is already in the form I need.So substituting back in:[ 4 cdot 5^{1002} cdot 6^{1001} = 4 cdot 5 cdot 5^{1001} cdot 6^{1001} = 20 cdot (5^{1001} cdot 6^{1001}) = 20 cdot 30^{1001} ]Wait, but the original equation has ( k cdot 30^{1001} ). So according to this, ( k ) should be 20. But hold on, let me double-check my steps because sometimes I might make a mistake in the exponents.Going back through the steps:1. I used the difference of squares correctly: ( a^2 - b^2 = (a - b)(a + b) ).2. Calculated ( a - b = 2 cdot 6^{1001} ) and ( a + b = 2 cdot 5^{1002} ). That seems right.3. Multiplied them together: ( 4 cdot 5^{1002} cdot 6^{1001} ). Correct.4. Then, expressed ( 30^{1001} ) as ( 5^{1001} cdot 6^{1001} ). That's accurate.5. Broke down ( 5^{1002} ) into ( 5 cdot 5^{1001} ). Yes, that's correct.So, substituting:[ 4 cdot 5^{1002} cdot 6^{1001} = 4 cdot 5 cdot 5^{1001} cdot 6^{1001} = 20 cdot 30^{1001} ]Hmm, so that would mean ( k = 20 ). But wait, the initial problem had exponents 1002 and 1001. Maybe I missed something with the exponents.Let me try another approach. Maybe instead of factoring, I can expand both squares and subtract. Let's see.First, expand ( (5^{1002} + 6^{1001})^2 ):[ (5^{1002})^2 + 2 cdot 5^{1002} cdot 6^{1001} + (6^{1001})^2 ]Which is:[ 5^{2004} + 2 cdot 5^{1002} cdot 6^{1001} + 6^{2002} ]Now, expand ( (5^{1002} - 6^{1001})^2 ):[ (5^{1002})^2 - 2 cdot 5^{1002} cdot 6^{1001} + (6^{1001})^2 ]Which is:[ 5^{2004} - 2 cdot 5^{1002} cdot 6^{1001} + 6^{2002} ]Subtracting the second expansion from the first:[ (5^{2004} + 2 cdot 5^{1002} cdot 6^{1001} + 6^{2002}) - (5^{2004} - 2 cdot 5^{1002} cdot 6^{1001} + 6^{2002}) ]Simplify term by term:- ( 5^{2004} - 5^{2004} = 0 )- ( 2 cdot 5^{1002} cdot 6^{1001} - (-2 cdot 5^{1002} cdot 6^{1001}) = 4 cdot 5^{1002} cdot 6^{1001} )- ( 6^{2002} - 6^{2002} = 0 )So, the result is ( 4 cdot 5^{1002} cdot 6^{1001} ), which matches what I got earlier.Now, to express this as ( k cdot 30^{1001} ), let's see:We have ( 4 cdot 5^{1002} cdot 6^{1001} ). Let's factor out ( 5^{1001} ) and ( 6^{1001} ):[ 4 cdot 5^{1001} cdot 5 cdot 6^{1001} = 4 cdot 5 cdot (5^{1001} cdot 6^{1001}) = 20 cdot 30^{1001} ]So, that still gives ( k = 20 ). Wait, but earlier I thought maybe I made a mistake, but both methods give me 20. Maybe I was overcomplicating it.But hold on, let me check the exponents again. The original expression is ( (5^{1002} + 6^{1001})^2 - (5^{1002} - 6^{1001})^2 ). So, the exponents are 1002 and 1001, which are different. Did I handle that correctly?Yes, because when I expanded, I kept the exponents as they were. So, ( 5^{1002} ) squared is ( 5^{2004} ), and ( 6^{1001} ) squared is ( 6^{2002} ). Then, when subtracting, those terms canceled out, leaving the middle terms.So, I think my calculation is correct, and ( k = 20 ). But wait, the initial problem had ( 30^{1001} ), and I have ( 30^{1001} ) in my final expression. So, 20 times that is correct.But just to be absolutely sure, let me try plugging in smaller exponents to see if the pattern holds. Let's say instead of 1002 and 1001, use 2 and 1.So, compute ( (5^2 + 6^1)^2 - (5^2 - 6^1)^2 ) and see if it equals ( k cdot 30^1 ).Calculate each part:First, ( 5^2 = 25 ), ( 6^1 = 6 ).Compute ( (25 + 6)^2 - (25 - 6)^2 ):- ( 31^2 = 961 )- ( 19^2 = 361 )- Difference: ( 961 - 361 = 600 )Now, ( 30^1 = 30 ), so ( k cdot 30 = 600 ) implies ( k = 20 ).So, in this smaller case, ( k = 20 ). That seems to confirm my result.Therefore, I think my answer is correct. ( k = 20 ).Wait, but in the initial problem, the exponents are 1002 and 1001, which are consecutive. So, maybe there's a pattern here where ( k ) is always 20 regardless of the exponents? Or does it depend on the exponents?Wait, in my small case, the exponents were 2 and 1, and ( k = 20 ). In the original problem, exponents are 1002 and 1001, so similar pattern. So, maybe it's consistent.But let me try another small exponent to check. Let's use exponents 3 and 2.Compute ( (5^3 + 6^2)^2 - (5^3 - 6^2)^2 ).First, ( 5^3 = 125 ), ( 6^2 = 36 ).Compute ( (125 + 36)^2 - (125 - 36)^2 ):- ( 161^2 = 25921 )- ( 89^2 = 7921 )- Difference: ( 25921 - 7921 = 18000 )Now, ( 30^2 = 900 ), so ( k cdot 900 = 18000 ) implies ( k = 20 ).Again, ( k = 20 ). So, it seems consistent. Therefore, I can be confident that ( k = 20 ) is the correct answer.But wait, in the initial problem, the exponent on 30 is 1001, which is one less than 1002. In my small examples, I used exponents like 2 and 1, 3 and 2, and the exponent on 30 was one less. So, that's consistent.Therefore, I think my answer is correct. ( k = 20 ).

💡Alright, so I've got this problem here about functions and their symmetry with respect to the line y = x. Let me try to wrap my head around it step by step.First, the problem says that the graph of y = g(x) is symmetric to the graph of f(x) = 2^x with respect to the line y = x. Hmm, okay. I remember that when two graphs are symmetric with respect to the line y = x, it usually means that they are inverses of each other. So, if f(x) is one function, then g(x) should be its inverse. That makes sense because reflecting a function over the line y = x gives you its inverse function.So, if f(x) = 2^x, then g(x) should be the inverse of f(x). To find the inverse, I need to switch the roles of x and y and solve for y. Let me write that down:Starting with y = 2^x.To find the inverse, swap x and y:x = 2^y.Now, I need to solve for y. Hmm, how do I do that? Well, I know that logarithms are the inverse operations of exponentials. So, if I take the logarithm base 2 of both sides, I can solve for y.Taking log base 2 of both sides:log₂(x) = y.So, that means the inverse function is y = log₂(x). Therefore, g(x) = log₂(x).Alright, so now I need to find the value of g(1/2). Let's plug in 1/2 into the function g(x):g(1/2) = log₂(1/2).Hmm, what's log base 2 of 1/2? I remember that log₂(1) is 0 because 2^0 = 1. And log₂(2) is 1 because 2^1 = 2. So, 1/2 is less than 1, which means the logarithm should be negative. Let me think about it more carefully.I know that 2^(-1) = 1/2. So, if 2 raised to the power of -1 equals 1/2, then log₂(1/2) must be -1. That makes sense because logarithms answer the question: "To what power should I raise the base to get this number?" So, in this case, the base is 2, and we want to get 1/2. The answer is -1.Therefore, g(1/2) = -1.Let me double-check my reasoning. If f(x) = 2^x, then its inverse should indeed be g(x) = log₂(x). Plugging in 1/2 into g(x) gives log₂(1/2), which is -1 because 2^(-1) = 1/2. That seems correct.I don't think I made any mistakes here. The key was recognizing that symmetry with respect to y = x implies that the functions are inverses of each other. Once I found the inverse function, plugging in 1/2 was straightforward.So, I'm pretty confident that the answer is -1, which corresponds to option D.**Final Answer**boxed{D}

💡Okay, so I need to prove that for a point P lying outside a circle S, its power with respect to S is equal to the square of the length of the tangent drawn from this point. Hmm, I remember that the power of a point has something to do with the product of the lengths of segments created by lines through the point intersecting the circle. But I'm not entirely sure about all the details. Let me try to recall and work through this step by step.First, let me visualize the situation. There's a circle S with center O and radius R. Point P is outside this circle. From P, I can draw a tangent to the circle, which touches the circle at point T. I also know that from P, I can draw a secant that intersects the circle at two points, say A and B. The power of point P with respect to circle S is defined as the product PA * PB. I need to show that this product is equal to PT squared, where PT is the length of the tangent from P to the circle.Alright, so maybe I should start by recalling the Power of a Point theorem. I think it states that for a point outside a circle, the product of the lengths of the segments of any secant line from that point is equal to the square of the length of the tangent from that point to the circle. So, in other words, PA * PB = PT^2. That sounds exactly like what I need to prove. But I need to actually prove this theorem, not just state it.Let me try to set up some variables and see if I can derive this. Let's denote the distance from P to the center O as d. Since P is outside the circle, d must be greater than the radius R. The tangent PT forms a right angle with the radius OT at the point of tangency T. So, triangle OPT is a right-angled triangle with the right angle at T.In triangle OPT, by the Pythagorean theorem, we have:OP^2 = OT^2 + PT^2But OT is the radius R, so:d^2 = R^2 + PT^2Therefore, PT^2 = d^2 - R^2Okay, so PT squared is equal to the square of the distance from P to the center minus the square of the radius. That's a useful expression.Now, let's consider the secant line passing through P and intersecting the circle at points A and B. Let's denote PA as the length from P to A and PB as the length from P to B. Since A and B are points on the circle, OA and OB are both radii of the circle, so OA = OB = R.I need to relate PA and PB to d and R. Maybe I can use the Power of a Point theorem here, but I'm supposed to prove it, not use it. So perhaps I should use coordinate geometry or some other method.Alternatively, maybe I can use similar triangles. Let me think. If I draw the tangent PT and the secant PAB, is there a way to relate these using similar triangles?Wait, I recall that the tangent-secant theorem states that the angle between the tangent and the chord (PT and TA, for example) is equal to the angle in the alternate segment. That might be useful, but I'm not sure how to apply it here.Let me try to express PA and PB in terms of d and R. Let's consider the distances PA and PB. Let me denote PA as x and PB as y. Since A and B are points on the circle, the distances from O to A and O to B are both R.Using the Law of Cosines in triangles OPA and OPB might help. Let's denote the angle at O between OP and OA as θ. Then, in triangle OPA:PA^2 = OP^2 + OA^2 - 2 * OP * OA * cosθSimilarly, in triangle OPB:PB^2 = OP^2 + OB^2 - 2 * OP * OB * cosφBut since OA and OB are both R, and OP is d, we have:PA^2 = d^2 + R^2 - 2dR cosθPB^2 = d^2 + R^2 - 2dR cosφBut I'm not sure if this is leading me anywhere. Maybe I need a different approach.Wait, let's think about the power of a point again. The power is defined as PA * PB. If I can express PA * PB in terms of d and R, and then show that it equals PT^2, which is d^2 - R^2, then I'm done.So, let's try to express PA * PB. Let me denote PA as x and PB as y. Then, PA * PB = x * y.But in the secant line, PA is the distance from P to A, and PB is the distance from P to B. Since A and B are on the circle, and P is outside, PA is the shorter segment and PB is the longer segment. So, PA = x and PB = x + AB.But I don't know AB. Maybe I can express AB in terms of R and the angle between OA and OB.Alternatively, maybe I can use the intersecting chords theorem. Wait, but that applies when two chords intersect inside the circle, not when one is a secant from outside.Hmm, I'm getting a bit stuck here. Let me try to think differently.I know that PT is the tangent, and PT^2 = d^2 - R^2. If I can express PA * PB in terms of d and R, and show that it's equal to d^2 - R^2, then I'm done.Let me consider the coordinates. Let's place the circle S with center at the origin (0,0) for simplicity. Let P be at (d, 0), so the distance from P to O is d. The tangent from P to the circle will touch the circle at some point T. Let's find the coordinates of T.The equation of the circle is x^2 + y^2 = R^2. The equation of the tangent from P(d, 0) to the circle can be found using the formula for the tangent from an external point. The equation of the tangent is:xx1 + yy1 = R^2Where (x1, y1) is the point of tangency. But since P is (d, 0), the tangent line from P will have the equation:x * d + y * 0 = R^2So, x = R^2 / dTherefore, the point T is at (R^2 / d, y). To find y, substitute x into the circle equation:(R^2 / d)^2 + y^2 = R^2So, y^2 = R^2 - (R^4 / d^2) = (d^2 R^2 - R^4) / d^2 = R^2 (d^2 - R^2) / d^2Therefore, y = ± R sqrt(d^2 - R^2) / dSo, the coordinates of T are (R^2 / d, R sqrt(d^2 - R^2) / d) and (R^2 / d, - R sqrt(d^2 - R^2) / d). But since we're dealing with lengths, we can take the positive value.Now, the length of the tangent PT is the distance from P(d, 0) to T(R^2 / d, R sqrt(d^2 - R^2) / d). Let's compute this distance.PT^2 = (d - R^2 / d)^2 + (0 - R sqrt(d^2 - R^2) / d)^2Let's compute each term:First term: (d - R^2 / d)^2 = ( (d^2 - R^2) / d )^2 = (d^2 - R^2)^2 / d^2Second term: ( - R sqrt(d^2 - R^2) / d )^2 = ( R^2 (d^2 - R^2) ) / d^2So, PT^2 = (d^2 - R^2)^2 / d^2 + R^2 (d^2 - R^2) / d^2Factor out (d^2 - R^2) / d^2:PT^2 = (d^2 - R^2) / d^2 [ (d^2 - R^2) + R^2 ] = (d^2 - R^2) / d^2 * d^2 = d^2 - R^2So, PT^2 = d^2 - R^2, which matches what we had earlier.Now, let's consider the secant line passing through P and intersecting the circle at A and B. Let's find the product PA * PB.Let me parameterize the secant line. Since P is at (d, 0), and the secant passes through P, let's assume it makes an angle θ with the x-axis. The parametric equations of the secant line are:x = d + t cosθy = 0 + t sinθWe can find the points A and B where this line intersects the circle x^2 + y^2 = R^2.Substitute x and y into the circle equation:(d + t cosθ)^2 + (t sinθ)^2 = R^2Expand this:d^2 + 2 d t cosθ + t^2 cos^2θ + t^2 sin^2θ = R^2Simplify:d^2 + 2 d t cosθ + t^2 (cos^2θ + sin^2θ) = R^2Since cos^2θ + sin^2θ = 1:d^2 + 2 d t cosθ + t^2 = R^2Rearrange:t^2 + 2 d t cosθ + (d^2 - R^2) = 0This is a quadratic equation in t. Let's denote the roots as t1 and t2, which correspond to the parameters for points A and B.From quadratic equation theory, the product of the roots t1 * t2 = (d^2 - R^2)But t represents the parameter along the secant line from P. So, the distances PA and PB correspond to |t1| and |t2|.However, since P is outside the circle, both t1 and t2 are positive (assuming the secant goes through the circle in the direction of θ). Therefore, PA = t1 and PB = t2, so PA * PB = t1 * t2 = d^2 - R^2But we also have PT^2 = d^2 - R^2, so PA * PB = PT^2Therefore, we've shown that the power of point P with respect to circle S, which is PA * PB, is equal to the square of the length of the tangent PT from P to S.So, the proof is complete.

💡First, I need to calculate the distance Joseph drives by multiplying his speed by the time he spends driving.Next, I'll calculate the distance Kyle drives using the same method.Finally, I'll find the difference between the distances driven by Joseph and Kyle to determine how many more miles Joseph drives than Kyle.

💡First, I recognize that the product of two numbers is equal to the product of their HCF and LCM. Given that the HCF is 84 and the LCM is 21, I can set up the equation:84 * B = 84 * 21Next, I solve for B by dividing both sides of the equation by 84:B = (84 * 21) / 84B = 21Now that I have both numbers, A = 84 and B = 21, I determine their ratio:A:B = 84:21To simplify the ratio, I divide both numbers by their greatest common divisor, which is 21:A:B = 4:1Therefore, the ratio of the two numbers is 4:1.

💡Alright, so I'm trying to figure out why a particular option would give the highest average income for the department employees. The problem mentions that if I choose any option, I need to justify why that option would result in the highest average income. First, I need to understand what's being asked. It seems like there are multiple options or methods for distributing income among employees, and I need to choose the one that maximizes the average income. But wait, the initial statement says "if you chose any option," which is a bit confusing. Does it mean that no matter which option I choose, the average income remains the same? Or does it mean that I need to choose a specific option that would lead to the highest average income?Looking back at the problem, it mentions that the average income does not depend on the distribution method. That suggests that no matter how the bonuses are distributed, the average income remains constant. So, if that's the case, then choosing any option wouldn't change the average income because it's determined by the total salary and total bonus, not how those bonuses are allocated among employees.But wait, the user is asking me to justify why a chosen option would give the highest average income. If the average income is independent of the distribution method, then perhaps the highest average income is already determined by the total salary and total bonus, and it doesn't matter how the bonuses are distributed. Let me think about this more carefully. If I have a fixed total salary and a fixed total bonus, then the average income is just the sum of these two divided by the number of employees. So, if I have 10 employees, the average income would be (Total Salary + Total Bonus) / 10. This doesn't change regardless of how the bonus is distributed among the employees. Therefore, if the total salary and total bonus are fixed, the average income is fixed as well, and it doesn't depend on how the bonuses are allocated. So, in that sense, choosing any distribution method wouldn't affect the average income because it's already determined by the totals.But the user is asking to justify why a chosen option would give the highest average income. If the average income is fixed, then maybe the question is more about ensuring that the distribution method doesn't inadvertently lower the average income. For example, if bonuses are distributed in a way that some employees receive significantly more than others, could that affect the average? Wait, no. The average is just the total divided by the number of employees, so individual distributions don't affect the average. So, regardless of how the bonuses are distributed, the average income remains the same. Therefore, any distribution method would result in the same average income, and there's no need to choose a specific option to maximize it because it's already maximized by the totals.But then why is the user asking to justify why a chosen option would give the highest average income? Maybe there's a misunderstanding here. Perhaps the user is thinking that different distribution methods could lead to different average incomes, but in reality, as long as the total salary and total bonus are fixed, the average income remains constant.Alternatively, maybe the user is considering different options that could affect the total bonus or total salary. For example, if one option increases the total bonus while keeping the total salary the same, that would increase the average income. In that case, choosing the option that increases the total bonus would indeed lead to a higher average income.But the initial problem statement doesn't specify that the total bonus or total salary can change. It just mentions that the average income doesn't depend on the distribution method. So, perhaps the user is conflating the distribution method with the total amounts.In summary, if the total salary and total bonus are fixed, then the average income is fixed, and it doesn't matter how the bonuses are distributed. Therefore, choosing any option wouldn't change the average income. However, if the options involve changing the total bonus or total salary, then choosing the option that increases these totals would lead to a higher average income.Given the problem statement, it seems like the focus is on the distribution method rather than changing the totals. Therefore, the average income remains the same regardless of the distribution method, and there's no need to choose a specific option to maximize it because it's already determined by the totals.But to fully address the user's question, I should consider both interpretations. If the goal is to maximize the average income, and the totals are fixed, then no distribution method will change the average. However, if the option allows for increasing the total bonus or salary, then choosing that option would indeed lead to a higher average income.In conclusion, to justify why a chosen option would give the highest average income, I need to consider whether the option affects the total salary or total bonus. If it does, then increasing these totals would directly increase the average income. If the option only changes the distribution method without affecting the totals, then the average income remains unchanged.