answer：Alright, so I have this problem here: Given the expansion of (2x - sqrt(3))^10, which is equal to a_0 + a_1x + a_2x^2 + ... + a_{10}x^{10}. There are two parts to the problem. First, I need to find a_0. Hmm, okay. I remember that in a polynomial expansion, the constant term, which is a_0 here, is the term where there's no x. So, how do I find that? Well, in the binomial expansion, each term is given by C(n, k) * (2x)^{n - k} * (-sqrt(3))^k. So, for a_0, which is the constant term, the exponent of x should be zero. That means (2x)^{10 - k} should have x^0, so 10 - k must be zero. Therefore, k = 10. So, substituting k = 10 into the term, we get C(10, 10)*(2x)^{0}*(-sqrt(3))^{10}. Simplifying that, C(10,10) is 1, (2x)^0 is 1, and (-sqrt(3))^{10}. Since the exponent is even, the negative sign will disappear, so it's (sqrt(3))^{10}. Now, sqrt(3) is 3^{1/2}, so (3^{1/2})^{10} is 3^{5}, which is 243. So, a_0 is 243. That seems straightforward.Okay, moving on to part (2). I need to find the product of two sums: (a_0 + a_1 + a_2 + ... + a_{10}) and (a_0 - a_1 + a_2 - a_3 + ... + a_{10}). Hmm, that looks like evaluating the polynomial at x = 1 and x = -1, and then multiplying those results together. Let me think. If I plug in x = 1 into the polynomial, I get (2*1 - sqrt(3))^{10} = (2 - sqrt(3))^{10}, which is exactly the sum a_0 + a_1 + a_2 + ... + a_{10}. Similarly, if I plug in x = -1, I get (2*(-1) - sqrt(3))^{10} = (-2 - sqrt(3))^{10}, which is the alternating sum a_0 - a_1 + a_2 - a_3 + ... + a_{10}. So, the product we're looking for is [(2 - sqrt(3))^{10}] * [(-2 - sqrt(3))^{10}]. Hmm, that's a bit complicated, but maybe I can simplify this expression. I notice that both terms are raised to the 10th power, so perhaps I can combine them first before raising to the power. Let me see: (2 - sqrt(3))*(-2 - sqrt(3)). Let's compute this product. Multiplying (2 - sqrt(3)) and (-2 - sqrt(3)): First, multiply 2 and -2: that's -4.Then, 2 and -sqrt(3): that's -2sqrt(3).Then, -sqrt(3) and -2: that's +2sqrt(3).Finally, -sqrt(3) and -sqrt(3): that's + (sqrt(3))^2, which is 3.So, adding all these up: -4 - 2sqrt(3) + 2sqrt(3) + 3. The -2sqrt(3) and +2sqrt(3) cancel each other out, leaving -4 + 3 = -1.Wow, okay, so (2 - sqrt(3))*(-2 - sqrt(3)) = -1. Therefore, [(2 - sqrt(3))*(-2 - sqrt(3))]^{10} = (-1)^{10} = 1.Wait, hold on. Because both (2 - sqrt(3)) and (-2 - sqrt(3)) are being raised to the 10th power, and then multiplied together. So, actually, it's [(2 - sqrt(3))^{10}] * [(-2 - sqrt(3))^{10}] = [(2 - sqrt(3))*(-2 - sqrt(3))]^{10} = (-1)^{10} = 1. So, the product is 1. That's interesting. So, the value of (a_0 + a_1 + ... + a_{10})(a_0 - a_1 + ... + a_{10}) is 1.Wait, let me double-check that. So, (2 - sqrt(3))^{10} * (-2 - sqrt(3))^{10} is equal to [(2 - sqrt(3))*(-2 - sqrt(3))]^{10}, which is (-1)^{10}, which is 1. That seems correct.Alternatively, I can think of it as (2 - sqrt(3))^{10} * (2 + sqrt(3))^{10} if I factor out the negative sign from (-2 - sqrt(3))^{10}, which would be [(-1)*(2 + sqrt(3))]^{10} = (-1)^{10}*(2 + sqrt(3))^{10} = (2 + sqrt(3))^{10}. So, then, (2 - sqrt(3))^{10}*(2 + sqrt(3))^{10} is [(2 - sqrt(3))(2 + sqrt(3))]^{10} = (4 - 3)^{10} = (1)^{10} = 1. Yeah, that also gives 1. So, that confirms it.So, both methods lead to the same result, which is 1. Therefore, the product is 1.So, to recap:1. a_0 is 243.2. The product of the two sums is 1.I think that's it. Let me just make sure I didn't make any calculation errors.For part (1), the constant term is when k = 10, so C(10,10)*(2x)^0*(-sqrt(3))^{10} = 1*1*(sqrt(3))^{10} = 3^5 = 243. Yep, that's correct.For part (2), evaluating at x = 1 and x = -1, multiplying, and simplifying gives 1. That seems solid.**Final Answer**(1) boxed{243}(2) boxed{1}

answer：Okay, so I have this quadratic equation: ( x^2 + x - a = 0 ). The problem says that one of the roots is 2, and I need to find the other root. Hmm, let me think about how to approach this.First, I remember that for any quadratic equation of the form ( ax^2 + bx + c = 0 ), the sum of the roots is ( -b/a ) and the product of the roots is ( c/a ). Wait, actually, in this case, the equation is ( x^2 + x - a = 0 ), so the coefficients are a bit different. Let me write that down:The general form is ( ax^2 + bx + c = 0 ), so in this case, ( a = 1 ), ( b = 1 ), and ( c = -a ). Hmm, that might be confusing because the constant term is also denoted by ( a ). Maybe I should use a different notation to avoid confusion. Let me rewrite the equation as ( x^2 + x - k = 0 ), where ( k ) is the constant term. So, in this case, ( a = 1 ), ( b = 1 ), and ( c = -k ).Wait, but the original equation is ( x^2 + x - a = 0 ), so actually, the constant term is ( -a ). Maybe I should just stick with that. So, in the standard form, ( a_{text{quad}} = 1 ), ( b_{text{quad}} = 1 ), and ( c_{text{quad}} = -a ).Given that, the sum of the roots is ( -b_{text{quad}} / a_{text{quad}} ) which is ( -1/1 = -1 ). And the product of the roots is ( c_{text{quad}} / a_{text{quad}} ) which is ( (-a)/1 = -a ).So, if one root is 2, let's denote the other root as ( r ). Then, according to the sum of roots:( 2 + r = -1 )So, solving for ( r ):( r = -1 - 2 = -3 )Wait, so the other root is -3? Let me check that.Alternatively, using the product of the roots:( 2 times r = -a )But I don't know the value of ( a ) yet. Maybe I can find ( a ) first using the fact that 2 is a root.If 2 is a root, then substituting ( x = 2 ) into the equation should satisfy it:( (2)^2 + 2 - a = 0 )Calculating that:( 4 + 2 - a = 0 )So, ( 6 - a = 0 ), which gives ( a = 6 ).Now, knowing that ( a = 6 ), the product of the roots is ( -a = -6 ). So, ( 2 times r = -6 ), which means ( r = -6 / 2 = -3 ). So, that confirms it.Alternatively, I could have just used the sum of the roots without finding ( a ). Since the sum is -1, and one root is 2, the other must be -3. Either way, I get the same result.Just to make sure, let me plug ( x = -3 ) into the equation and see if it satisfies it with ( a = 6 ):( (-3)^2 + (-3) - 6 = 0 )Calculating:( 9 - 3 - 6 = 0 )Which simplifies to ( 0 = 0 ). Perfect, that works.So, both methods give me the same answer, which is reassuring. I think I'm confident that the other root is -3.**Final Answer**The other root is boxed{-3}.

answer：First, I need to determine the total distance Barney traveled. The odometer changed from 1441 miles to 1661 miles, so the distance ridden is 1661 - 1441 = 220 miles.Next, I'll calculate the total time he spent riding. He rode for 4 hours on the first day and 6 hours on the next day, which adds up to 4 + 6 = 10 hours.Finally, to find the average speed, I'll divide the total distance by the total time. That gives 220 miles divided by 10 hours, resulting in an average speed of 22 miles per hour.

answer：Okay, so I have this problem here where I need to figure out how many different results we can get from the expression:frac{a}{|a|} + frac{b}{|b|} + frac{c}{|c|} + frac{abc}{|abc|}where (a), (b), and (c) are non-zero real numbers. Hmm, let me break this down step by step.First, I remember that for any non-zero real number (x), the expression (frac{x}{|x|}) is equal to 1 if (x) is positive and -1 if (x) is negative. So, each of these terms (frac{a}{|a|}), (frac{b}{|b|}), and (frac{c}{|c|}) can only be either 1 or -1. That simplifies things a bit because instead of dealing with all real numbers, I can just consider the signs of (a), (b), and (c).Let me denote:- (s_a = frac{a}{|a|})- (s_b = frac{b}{|b|})- (s_c = frac{c}{|c|})So, each (s_a), (s_b), (s_c) is either 1 or -1. Then, the expression becomes:s_a + s_b + s_c + frac{abc}{|abc|}But wait, (frac{abc}{|abc|}) is just the product of (s_a), (s_b), and (s_c). Because:frac{abc}{|abc|} = frac{a}{|a|} cdot frac{b}{|b|} cdot frac{c}{|c|} = s_a s_b s_cSo, substituting that in, the entire expression simplifies to:s_a + s_b + s_c + s_a s_b s_cAlright, so now the problem reduces to finding how many different values the expression (s_a + s_b + s_c + s_a s_b s_c) can take, where each (s_a), (s_b), (s_c) is either 1 or -1.Since each variable can be either 1 or -1, there are (2^3 = 8) possible combinations of (s_a), (s_b), and (s_c). Maybe I can list all of them and compute the expression for each case. That should give me all possible results, and then I can count the distinct ones.Let me list all 8 possibilities:1. (s_a = 1), (s_b = 1), (s_c = 1)2. (s_a = 1), (s_b = 1), (s_c = -1)3. (s_a = 1), (s_b = -1), (s_c = 1)4. (s_a = 1), (s_b = -1), (s_c = -1)5. (s_a = -1), (s_b = 1), (s_c = 1)6. (s_a = -1), (s_b = 1), (s_c = -1)7. (s_a = -1), (s_b = -1), (s_c = 1)8. (s_a = -1), (s_b = -1), (s_c = -1)Now, let's compute the expression for each case.1. **Case 1: All 1s** (s_a = 1), (s_b = 1), (s_c = 1) Expression: (1 + 1 + 1 + (1 times 1 times 1) = 1 + 1 + 1 + 1 = 4)2. **Case 2: Two 1s, one -1** (s_a = 1), (s_b = 1), (s_c = -1) Expression: (1 + 1 + (-1) + (1 times 1 times -1) = 1 + 1 - 1 - 1 = 0)3. **Case 3: Two 1s, one -1** (s_a = 1), (s_b = -1), (s_c = 1) Expression: (1 + (-1) + 1 + (1 times -1 times 1) = 1 - 1 + 1 - 1 = 0)4. **Case 4: One 1, two -1s** (s_a = 1), (s_b = -1), (s_c = -1) Expression: (1 + (-1) + (-1) + (1 times -1 times -1) = 1 - 1 - 1 + 1 = 0)5. **Case 5: Two 1s, one -1** (s_a = -1), (s_b = 1), (s_c = 1) Expression: (-1 + 1 + 1 + (-1 times 1 times 1) = -1 + 1 + 1 - 1 = 0)6. **Case 6: One 1, two -1s** (s_a = -1), (s_b = 1), (s_c = -1) Expression: (-1 + 1 + (-1) + (-1 times 1 times -1) = -1 + 1 - 1 + 1 = 0)7. **Case 7: One 1, two -1s** (s_a = -1), (s_b = -1), (s_c = 1) Expression: (-1 + (-1) + 1 + (-1 times -1 times 1) = -1 - 1 + 1 + 1 = 0)8. **Case 8: All -1s** (s_a = -1), (s_b = -1), (s_c = -1) Expression: (-1 + (-1) + (-1) + (-1 times -1 times -1) = -1 -1 -1 -1 = -4)So, compiling the results:- Case 1: 4- Cases 2-7: 0- Case 8: -4Wait, so all the cases except the first and the last result in 0. So, the possible results are 4, 0, and -4. That would mean there are three different results.But hold on, let me double-check my calculations because that seems a bit too straightforward. Maybe I made a mistake in computing some of the cases.Let me go through each case again.**Case 1: All 1s**1 + 1 + 1 + 1 = 4. That seems correct.**Case 2: s_a=1, s_b=1, s_c=-1**1 + 1 -1 + (1*1*(-1)) = 1 + 1 -1 -1 = 0. Correct.**Case 3: s_a=1, s_b=-1, s_c=1**1 -1 +1 + (1*(-1)*1) = 1 -1 +1 -1 = 0. Correct.**Case 4: s_a=1, s_b=-1, s_c=-1**1 -1 -1 + (1*(-1)*(-1)) = 1 -1 -1 +1 = 0. Correct.**Case 5: s_a=-1, s_b=1, s_c=1**-1 +1 +1 + (-1*1*1) = -1 +1 +1 -1 = 0. Correct.**Case 6: s_a=-1, s_b=1, s_c=-1**-1 +1 -1 + (-1*1*(-1)) = -1 +1 -1 +1 = 0. Correct.**Case 7: s_a=-1, s_b=-1, s_c=1**-1 -1 +1 + (-1*(-1)*1) = -1 -1 +1 +1 = 0. Correct.**Case 8: All -1s**-1 -1 -1 + (-1*(-1)*(-1)) = -1 -1 -1 -1 = -4. Correct.So, all the cases check out. So, the expression can be 4, 0, or -4. Therefore, there are 3 different results.But wait, let me think again. Is there a way that the expression can result in other numbers? For example, could it be 2 or -2?Wait, maybe I should consider another approach. Let me think about the expression (s_a + s_b + s_c + s_a s_b s_c). Let me denote (S = s_a + s_b + s_c) and (P = s_a s_b s_c). So, the expression is (S + P).Now, since each (s) is either 1 or -1, (S) can range from -3 to 3 in steps of 2 (since each term is odd). So possible values for (S) are -3, -1, 1, 3.Similarly, (P) is the product of three 1s or -1s, so (P) is 1 if there are an even number of -1s and -1 if there are an odd number of -1s. Wait, no: actually, the product of three numbers each being 1 or -1 is 1 if there are an even number of -1s and -1 if there are an odd number of -1s. Because (-1)^even = 1 and (-1)^odd = -1.So, if the number of -1s among (s_a), (s_b), (s_c) is even, (P = 1); if odd, (P = -1).Therefore, let's see:- If (S = 3) (all 1s), then (P = 1). So, (S + P = 4).- If (S = 1), which occurs when two are 1 and one is -1, then (P = -1) (since one -1 is odd). So, (S + P = 1 - 1 = 0).- If (S = -1), which occurs when two are -1 and one is 1, then (P = -1) (since two -1s is even? Wait, hold on: two -1s is even, so (P = 1). Wait, no: the product is (-1)^2 * 1 = 1. So, if two are -1s, (P = 1). So, (S + P = -1 + 1 = 0).- If (S = -3) (all -1s), then (P = (-1)^3 = -1). So, (S + P = -3 -1 = -4).So, regardless of whether (S) is 1 or -1, (S + P) is 0. So, the possible results are 4, 0, -4. So, three different results.Therefore, the number of different results we can obtain is 3.Wait, but in my initial calculation, all the cases with one -1 or two -1s gave 0, which matches this reasoning. So, yes, 3 different results.But just to make sure, let me think if there's another way this could result in a different value. Suppose, for example, two of the variables are 1 and one is -1. Then, (S = 1 + 1 -1 = 1), and (P = 1*1*(-1) = -1). So, (S + P = 0). Similarly, if two are -1 and one is 1, (S = -1 -1 +1 = -1), (P = (-1)*(-1)*1 = 1), so (S + P = 0). So, yeah, all those cases give 0.Therefore, the only distinct results are 4, 0, and -4. So, three different results.I think that's solid. So, the answer should be 3.**Final Answer**The number of different results we can obtain is boxed{3}.

answer：Alright, so I have this logarithm problem here: If log base 25 of (x minus 4) equals 1/2, I need to find 1 divided by log base x of 3. Hmm, okay, let's break this down step by step.First, I remember that logarithms can be converted into exponential form. The equation log_{25}(x - 4) = 1/2 means that 25 raised to the power of 1/2 equals x - 4. Let me write that down:25^(1/2) = x - 4.Now, 25 is 5 squared, so 25^(1/2) is the square root of 25, which is 5. So, 5 = x - 4. To find x, I just add 4 to both sides:5 + 4 = x,x = 9.Okay, so x is 9. Now, the next part is to find 1 divided by log base x of 3. Since x is 9, that becomes 1 divided by log base 9 of 3. Let me write that as:1 / log_9(3).Hmm, I need to figure out what log base 9 of 3 is. I remember that logarithms can be converted using the change of base formula, which is log_b(a) = log_c(a) / log_c(b) for any positive c, not equal to 1. Maybe I can use base 3 since 9 is a power of 3.Let me try that. So, log_9(3) can be written as log_3(3) / log_3(9). I know that log_3(3) is 1 because any log base a of a is 1. And log_3(9) is the exponent you need to raise 3 to get 9. Since 3 squared is 9, that's 2. So, log_3(9) is 2.Therefore, log_9(3) = 1 / 2. So, 1 divided by log_9(3) is 1 divided by (1/2), which is 2. So, the answer should be 2.Wait, let me double-check to make sure I didn't make any mistakes. Starting from the beginning, log_{25}(x - 4) = 1/2. Converting that to exponential form, 25^(1/2) = x - 4. 25^(1/2) is 5, so x is 9. That seems right.Then, 1 / log_9(3). Using change of base to base 3: log_9(3) = log_3(3) / log_3(9) = 1 / 2. So, 1 divided by (1/2) is indeed 2. Okay, that seems solid.Alternatively, I can think about log_9(3) as the exponent that 9 must be raised to in order to get 3. Since 9 is 3 squared, 9^(1/2) is 3. So, log_9(3) is 1/2, which confirms the earlier result.Therefore, 1 divided by log_9(3) is 2. I think that's correct.**Final Answer**The value of frac{1}{log_{x}3} is boxed{2}.

answer：Okay, so I have this algebra problem to solve. It says I need to simplify the expression first and then evaluate it, given some conditions. The expression is:(3/2)x²y - [x²y - 3(2xy - x²y) - xy]And the condition is (x - 1)² + |y + 2| = 0.Hmm, let me break this down step by step. First, I think I should simplify the expression before plugging in any values. So, let's focus on simplifying the expression.The expression is:(3/2)x²y - [x²y - 3(2xy - x²y) - xy]I need to simplify this. Let me start by looking at the part inside the brackets: [x²y - 3(2xy - x²y) - xy]. I'll simplify that first.So, inside the brackets, we have:x²y - 3(2xy - x²y) - xyLet me distribute the -3 across the terms inside the parentheses. That would be:x²y - 3*2xy + 3*x²y - xyCalculating each term:-3*2xy is -6xy-3*(-x²y) is +3x²ySo, substituting back, we have:x²y - 6xy + 3x²y - xyNow, let's combine like terms. The terms with x²y are x²y and 3x²y. That's 1x²y + 3x²y = 4x²y.The terms with xy are -6xy and -xy. That's -6xy -1xy = -7xy.So, the expression inside the brackets simplifies to:4x²y - 7xyNow, going back to the original expression:(3/2)x²y - [4x²y - 7xy]This becomes:(3/2)x²y - 4x²y + 7xyNow, let's combine the x²y terms. We have (3/2)x²y - 4x²y. To subtract these, it's easier if they have the same denominator.4x²y is the same as (8/2)x²y. So, (3/2)x²y - (8/2)x²y = (-5/2)x²y.So, now the expression is:(-5/2)x²y + 7xyI can factor out an xy from both terms:xy*(-5/2 x + 7)But maybe it's better to leave it as is for now. So, the simplified expression is:(-5/2)x²y + 7xyOkay, so that's the simplified form. Now, I need to evaluate this expression given the condition:(x - 1)² + |y + 2| = 0Hmm, this is interesting. Both (x - 1)² and |y + 2| are non-negative terms because squares and absolute values can't be negative. So, their sum is zero only if each term is zero individually.So, (x - 1)² = 0 and |y + 2| = 0.Solving these:For (x - 1)² = 0, take the square root of both sides: x - 1 = 0, so x = 1.For |y + 2| = 0, the absolute value is zero only when the inside is zero: y + 2 = 0, so y = -2.So, x = 1 and y = -2.Now, substitute these values into our simplified expression:(-5/2)x²y + 7xyLet's compute each term step by step.First, compute x²: x = 1, so x² = 1² = 1.Then, compute each term:First term: (-5/2)x²y = (-5/2)*1*(-2)Let me compute that:(-5/2)*1 = -5/2Then, multiply by (-2):-5/2 * (-2) = (5/2)*2 = 5So, the first term is 5.Second term: 7xy = 7*1*(-2) = 7*(-2) = -14So, adding the two terms together:5 + (-14) = 5 - 14 = -9So, the value of the expression is -9.Wait, let me double-check my calculations to make sure I didn't make a mistake.First term: (-5/2)x²yx = 1, y = -2x² = 1, so:(-5/2)*1*(-2) = (-5/2)*(-2) = (5/2)*2 = 5. That seems correct.Second term: 7xy = 7*1*(-2) = -14. That also seems correct.Adding them: 5 + (-14) = -9. Yep, that's right.So, after simplifying the expression and substituting x = 1 and y = -2, the value is -9.I think that's the correct answer. Let me just recap the steps to make sure I didn't skip anything.1. Simplified the expression step by step, distributing and combining like terms.2. Found that the simplified expression was (-5/2)x²y + 7xy.3. Solved the given condition to find x = 1 and y = -2.4. Substituted these values into the simplified expression.5. Calculated each term carefully and added them together.Everything seems to check out. So, I feel confident that the answer is -9.**Final Answer**The value of the expression is boxed{-9}.