answer：Alright, so I have this problem about an ellipse, and I need to solve two parts. Let me start by understanding what's given and what I need to find.First, the ellipse is given by the equation ( C: frac{x^2}{a^2} + frac{y^2}{b^2} = 1 ) where ( a > b > 0 ). The eccentricity of the ellipse is ( frac{sqrt{3}}{2} ), and there's a point ( left( -sqrt{3}, frac{1}{2} right) ) that lies on the ellipse. **Part (1): Find the standard equation of ellipse ( C ).**Okay, so I need to find the values of ( a ) and ( b ) to write the standard equation. Let me recall that the eccentricity ( e ) of an ellipse is given by ( e = frac{c}{a} ), where ( c ) is the distance from the center to each focus. Also, for an ellipse, ( c^2 = a^2 - b^2 ).Given that ( e = frac{sqrt{3}}{2} ), so ( frac{c}{a} = frac{sqrt{3}}{2} ). Therefore, ( c = frac{sqrt{3}}{2} a ).Now, since ( c^2 = a^2 - b^2 ), substituting ( c ) gives:( left( frac{sqrt{3}}{2} a right)^2 = a^2 - b^2 )Simplify the left side:( frac{3}{4} a^2 = a^2 - b^2 )Let me subtract ( frac{3}{4} a^2 ) from both sides:( 0 = a^2 - b^2 - frac{3}{4} a^2 )Simplify:( 0 = frac{1}{4} a^2 - b^2 )So, ( b^2 = frac{1}{4} a^2 ) or ( b = frac{a}{2} ).Alright, so now I have a relationship between ( a ) and ( b ). But I also know that the point ( left( -sqrt{3}, frac{1}{2} right) ) lies on the ellipse. So, plugging this point into the ellipse equation should satisfy it.Substitute ( x = -sqrt{3} ) and ( y = frac{1}{2} ) into ( frac{x^2}{a^2} + frac{y^2}{b^2} = 1 ):( frac{(-sqrt{3})^2}{a^2} + frac{left( frac{1}{2} right)^2}{b^2} = 1 )Simplify:( frac{3}{a^2} + frac{frac{1}{4}}{b^2} = 1 )But from earlier, ( b^2 = frac{a^2}{4} ), so substitute ( b^2 ):( frac{3}{a^2} + frac{frac{1}{4}}{frac{a^2}{4}} = 1 )Simplify the second term:( frac{frac{1}{4}}{frac{a^2}{4}} = frac{1}{a^2} )So now the equation becomes:( frac{3}{a^2} + frac{1}{a^2} = 1 )Combine the terms:( frac{4}{a^2} = 1 )Multiply both sides by ( a^2 ):( 4 = a^2 )So, ( a^2 = 4 ), which means ( a = 2 ) (since ( a > 0 )).Then, ( b^2 = frac{a^2}{4} = frac{4}{4} = 1 ), so ( b = 1 ).Therefore, the standard equation of the ellipse is:( frac{x^2}{4} + y^2 = 1 )That should be part (1) done.**Part (2): Find the maximum area of triangle ( POQ ).**Hmm, okay. So, we have a line ( l ) intersecting the ellipse at points ( P ) and ( Q ). The midpoint of segment ( PQ ) is ( H ), and the distance from the origin ( O ) to ( H ) is 1. We need to find the maximum area of triangle ( POQ ).Let me try to visualize this. The ellipse is ( frac{x^2}{4} + y^2 = 1 ), which is centered at the origin, stretching along the x-axis with semi-major axis 2 and semi-minor axis 1.A line intersects the ellipse at two points ( P ) and ( Q ). The midpoint ( H ) of ( PQ ) is such that the distance from the origin to ( H ) is 1. So, ( OH = 1 ).We need to find the maximum area of triangle ( POQ ). The area of triangle ( POQ ) can be found using the determinant formula:( text{Area} = frac{1}{2} |x_1 y_2 - x_2 y_1| )Alternatively, since ( P ) and ( Q ) are points on the ellipse, and ( O ) is the origin, the area can also be expressed in terms of vectors or coordinates.But maybe it's better to approach this using parametric equations or by considering the line equation and its intersection with the ellipse.Let me consider the general equation of a line. Let's suppose the line ( l ) has a slope ( m ) and intersects the x-axis at some point ( D(n, 0) ). So, the equation of the line can be written as ( x = m y + n ).Wait, or maybe it's better to write it in the form ( y = m x + c ). But since it intersects the x-axis at ( D(n, 0) ), the equation can be written as ( y = m(x - n) ). Hmm, but maybe using ( x = m y + n ) is better because it avoids issues when the line is vertical.Alternatively, parametric equations might be helpful, but perhaps I can proceed with substitution.Let me try substituting ( x = m y + n ) into the ellipse equation.So, substituting into ( frac{x^2}{4} + y^2 = 1 ):( frac{(m y + n)^2}{4} + y^2 = 1 )Expanding the numerator:( frac{m^2 y^2 + 2 m n y + n^2}{4} + y^2 = 1 )Multiply through by 4 to eliminate denominators:( m^2 y^2 + 2 m n y + n^2 + 4 y^2 = 4 )Combine like terms:( (m^2 + 4) y^2 + 2 m n y + (n^2 - 4) = 0 )This is a quadratic in ( y ). Let me denote this as:( A y^2 + B y + C = 0 ), where ( A = m^2 + 4 ), ( B = 2 m n ), and ( C = n^2 - 4 ).Let ( y_1 ) and ( y_2 ) be the roots of this quadratic, corresponding to the y-coordinates of points ( P ) and ( Q ).Then, the sum of the roots is ( y_1 + y_2 = -B/A = -2 m n / (m^2 + 4) ).Similarly, the product of the roots is ( y_1 y_2 = C/A = (n^2 - 4)/(m^2 + 4) ).Now, the midpoint ( H ) of segment ( PQ ) has coordinates:( H_x = frac{x_1 + x_2}{2} ), ( H_y = frac{y_1 + y_2}{2} ).But since ( x = m y + n ), we have:( x_1 = m y_1 + n ), ( x_2 = m y_2 + n ).Therefore,( H_x = frac{m y_1 + n + m y_2 + n}{2} = frac{m(y_1 + y_2) + 2n}{2} ).We already know ( y_1 + y_2 = -2 m n / (m^2 + 4) ), so:( H_x = frac{m(-2 m n / (m^2 + 4)) + 2n}{2} ).Simplify:( H_x = frac{ -2 m^2 n / (m^2 + 4) + 2n }{2} = frac{2n ( -m^2 / (m^2 + 4) + 1 ) }{2} ).Simplify inside the brackets:( -m^2 / (m^2 + 4) + 1 = ( -m^2 + m^2 + 4 ) / (m^2 + 4 ) = 4 / (m^2 + 4 ) ).Therefore,( H_x = frac{2n * 4 / (m^2 + 4)}{2} = frac{4n}{m^2 + 4} ).Similarly, ( H_y = (y_1 + y_2)/2 = (-2 m n / (m^2 + 4)) / 2 = - m n / (m^2 + 4) ).So, the coordinates of ( H ) are:( H left( frac{4n}{m^2 + 4}, - frac{m n}{m^2 + 4} right) ).Given that ( OH = 1 ), the distance from the origin to ( H ) is 1.So,( sqrt{ left( frac{4n}{m^2 + 4} right)^2 + left( - frac{m n}{m^2 + 4} right)^2 } = 1 ).Simplify inside the square root:( frac{16 n^2}{(m^2 + 4)^2} + frac{m^2 n^2}{(m^2 + 4)^2} = frac{(16 + m^2) n^2}{(m^2 + 4)^2} ).So,( sqrt{ frac{(16 + m^2) n^2}{(m^2 + 4)^2} } = 1 ).Simplify the square root:( frac{ sqrt{(16 + m^2)} |n| }{ m^2 + 4 } = 1 ).Therefore,( sqrt{16 + m^2} |n| = m^2 + 4 ).Let me square both sides to eliminate the square root:( (16 + m^2) n^2 = (m^2 + 4)^2 ).So,( n^2 = frac{(m^2 + 4)^2}{16 + m^2} ).Alright, so now I have an expression for ( n^2 ) in terms of ( m^2 ).Now, I need to find the area of triangle ( POQ ). Let me recall that the area can be given by:( text{Area} = frac{1}{2} | vec{OP} times vec{OQ} | ).But since ( P ) and ( Q ) are points on the plane, the area can also be expressed as:( text{Area} = frac{1}{2} |x_1 y_2 - x_2 y_1| ).Alternatively, since both points lie on the line ( l ), which is ( x = m y + n ), maybe I can express the area in terms of ( y_1 ) and ( y_2 ).Let me try that.Given ( x_1 = m y_1 + n ) and ( x_2 = m y_2 + n ), then:( x_1 y_2 - x_2 y_1 = (m y_1 + n) y_2 - (m y_2 + n) y_1 = m y_1 y_2 + n y_2 - m y_1 y_2 - n y_1 = n (y_2 - y_1) ).So, the area becomes:( text{Area} = frac{1}{2} |n (y_2 - y_1)| = frac{1}{2} |n| |y_2 - y_1| ).So, ( S = frac{1}{2} |n| |y_2 - y_1| ).Now, ( |y_2 - y_1| ) can be found from the quadratic equation. For a quadratic ( A y^2 + B y + C = 0 ), the difference of roots is ( sqrt{(y_1 + y_2)^2 - 4 y_1 y_2} ).So,( |y_2 - y_1| = sqrt{(y_1 + y_2)^2 - 4 y_1 y_2} ).From earlier, ( y_1 + y_2 = -2 m n / (m^2 + 4) ) and ( y_1 y_2 = (n^2 - 4)/(m^2 + 4) ).So,( |y_2 - y_1| = sqrt{ left( frac{-2 m n}{m^2 + 4} right)^2 - 4 cdot frac{n^2 - 4}{m^2 + 4} } ).Simplify inside the square root:First term: ( frac{4 m^2 n^2}{(m^2 + 4)^2} ).Second term: ( frac{4 (n^2 - 4)}{m^2 + 4} ).So,( |y_2 - y_1| = sqrt{ frac{4 m^2 n^2}{(m^2 + 4)^2} - frac{4 (n^2 - 4)}{m^2 + 4} } ).Let me factor out ( frac{4}{(m^2 + 4)^2} ):( |y_2 - y_1| = sqrt{ frac{4}{(m^2 + 4)^2} [ m^2 n^2 - (n^2 - 4)(m^2 + 4) ] } ).Simplify the expression inside the brackets:( m^2 n^2 - (n^2 - 4)(m^2 + 4) ).Expand ( (n^2 - 4)(m^2 + 4) ):( n^2 m^2 + 4 n^2 - 4 m^2 - 16 ).So,( m^2 n^2 - (n^2 m^2 + 4 n^2 - 4 m^2 - 16 ) = m^2 n^2 - n^2 m^2 - 4 n^2 + 4 m^2 + 16 = -4 n^2 + 4 m^2 + 16 ).Factor out 4:( 4 ( -n^2 + m^2 + 4 ) ).So, putting it back:( |y_2 - y_1| = sqrt{ frac{4}{(m^2 + 4)^2} cdot 4 ( -n^2 + m^2 + 4 ) } = sqrt{ frac{16 ( -n^2 + m^2 + 4 ) }{(m^2 + 4)^2} } ).Simplify the square root:( |y_2 - y_1| = frac{4 sqrt{ -n^2 + m^2 + 4 }}{m^2 + 4} ).So, the area ( S ) is:( S = frac{1}{2} |n| cdot frac{4 sqrt{ -n^2 + m^2 + 4 }}{m^2 + 4} = frac{2 |n| sqrt{ -n^2 + m^2 + 4 }}{m^2 + 4} ).Now, from earlier, we have ( n^2 = frac{(m^2 + 4)^2}{16 + m^2} ).Let me substitute ( n^2 ) into the expression for ( S ):First, compute ( -n^2 + m^2 + 4 ):( - frac{(m^2 + 4)^2}{16 + m^2} + m^2 + 4 ).Let me write ( m^2 + 4 ) as ( frac{(m^2 + 4)(16 + m^2)}{16 + m^2} ) to have a common denominator.So,( - frac{(m^2 + 4)^2}{16 + m^2} + frac{(m^2 + 4)(16 + m^2)}{16 + m^2} ).Combine the terms:( frac{ - (m^2 + 4)^2 + (m^2 + 4)(16 + m^2) }{16 + m^2} ).Factor out ( (m^2 + 4) ):( frac{(m^2 + 4) [ - (m^2 + 4) + (16 + m^2) ] }{16 + m^2} ).Simplify inside the brackets:( - (m^2 + 4) + (16 + m^2) = -m^2 - 4 + 16 + m^2 = 12 ).So,( frac{(m^2 + 4) cdot 12 }{16 + m^2} = frac{12 (m^2 + 4)}{16 + m^2} ).Therefore, ( -n^2 + m^2 + 4 = frac{12 (m^2 + 4)}{16 + m^2} ).So, going back to ( S ):( S = frac{2 |n| sqrt{ frac{12 (m^2 + 4)}{16 + m^2} } }{m^2 + 4} ).Simplify the square root:( sqrt{ frac{12 (m^2 + 4)}{16 + m^2} } = sqrt{12} cdot sqrt{ frac{m^2 + 4}{16 + m^2} } = 2 sqrt{3} cdot sqrt{ frac{m^2 + 4}{16 + m^2} } ).So,( S = frac{2 |n| cdot 2 sqrt{3} cdot sqrt{ frac{m^2 + 4}{16 + m^2} } }{m^2 + 4} = frac{4 sqrt{3} |n| sqrt{ frac{m^2 + 4}{16 + m^2} } }{m^2 + 4} ).Simplify:( S = frac{4 sqrt{3} |n| }{ sqrt{16 + m^2} } cdot frac{ sqrt{m^2 + 4} }{m^2 + 4} ).Wait, let me double-check that step.Wait, ( sqrt{ frac{m^2 + 4}{16 + m^2} } = frac{ sqrt{m^2 + 4} }{ sqrt{16 + m^2} } ).So,( S = frac{4 sqrt{3} |n| }{ m^2 + 4 } cdot frac{ sqrt{m^2 + 4} }{ sqrt{16 + m^2} } ).Wait, no, actually:Wait, ( S = frac{4 sqrt{3} |n| }{ m^2 + 4 } cdot sqrt{ frac{m^2 + 4}{16 + m^2} } ).Which is:( S = frac{4 sqrt{3} |n| }{ m^2 + 4 } cdot frac{ sqrt{m^2 + 4} }{ sqrt{16 + m^2} } = frac{4 sqrt{3} |n| }{ sqrt{16 + m^2} cdot sqrt{m^2 + 4} } cdot sqrt{m^2 + 4} ).Wait, that seems conflicting. Maybe I made a miscalculation.Wait, let me re-express ( S ):( S = frac{2 |n| sqrt{ frac{12 (m^2 + 4)}{16 + m^2} } }{m^2 + 4} ).So,( S = frac{2 |n| cdot sqrt{12} cdot sqrt{ frac{m^2 + 4}{16 + m^2} } }{m^2 + 4} ).Simplify ( sqrt{12} = 2 sqrt{3} ):( S = frac{2 |n| cdot 2 sqrt{3} cdot sqrt{ frac{m^2 + 4}{16 + m^2} } }{m^2 + 4} = frac{4 sqrt{3} |n| cdot sqrt{ frac{m^2 + 4}{16 + m^2} } }{m^2 + 4} ).Now, let me write ( sqrt{ frac{m^2 + 4}{16 + m^2} } = frac{ sqrt{m^2 + 4} }{ sqrt{16 + m^2} } ).So,( S = frac{4 sqrt{3} |n| cdot sqrt{m^2 + 4} }{ (m^2 + 4) sqrt{16 + m^2} } = frac{4 sqrt{3} |n| }{ sqrt{16 + m^2} cdot sqrt{m^2 + 4} } ).Wait, that seems a bit messy. Maybe I can express ( |n| ) in terms of ( m^2 ) using the earlier relation.We have ( n^2 = frac{(m^2 + 4)^2}{16 + m^2} ), so ( |n| = frac{m^2 + 4}{sqrt{16 + m^2}} ).Therefore, ( |n| = frac{m^2 + 4}{sqrt{16 + m^2}} ).Substituting back into ( S ):( S = frac{4 sqrt{3} cdot frac{m^2 + 4}{sqrt{16 + m^2}} }{ sqrt{16 + m^2} cdot sqrt{m^2 + 4} } = frac{4 sqrt{3} (m^2 + 4) }{ (16 + m^2) cdot sqrt{m^2 + 4} } ).Simplify:( S = frac{4 sqrt{3} sqrt{m^2 + 4} }{16 + m^2} ).So, ( S = frac{4 sqrt{3} sqrt{m^2 + 4} }{16 + m^2} ).Let me denote ( t = m^2 + 4 ). Since ( m^2 geq 0 ), ( t geq 4 ).So, ( S = frac{4 sqrt{3} sqrt{t} }{t + 12} ).Wait, because ( 16 + m^2 = (m^2 + 4) + 12 = t + 12 ).So, ( S = frac{4 sqrt{3} sqrt{t} }{t + 12} ).Now, I need to find the maximum value of ( S ) with respect to ( t geq 4 ).Let me set ( f(t) = frac{4 sqrt{3} sqrt{t} }{t + 12} ).To find the maximum, take the derivative of ( f(t) ) with respect to ( t ) and set it to zero.First, write ( f(t) = 4 sqrt{3} cdot t^{1/2} (t + 12)^{-1} ).Compute the derivative ( f'(t) ):Using the product rule:( f'(t) = 4 sqrt{3} left[ frac{1}{2} t^{-1/2} (t + 12)^{-1} + t^{1/2} (-1) (t + 12)^{-2} cdot 1 right] ).Simplify:( f'(t) = 4 sqrt{3} left[ frac{1}{2} cdot frac{1}{sqrt{t} (t + 12)} - frac{sqrt{t}}{(t + 12)^2} right] ).Factor out ( frac{1}{sqrt{t} (t + 12)^2} ):( f'(t) = 4 sqrt{3} cdot frac{1}{sqrt{t} (t + 12)^2} left[ frac{1}{2} (t + 12) - t right] ).Simplify inside the brackets:( frac{1}{2} (t + 12) - t = frac{t}{2} + 6 - t = - frac{t}{2} + 6 ).So,( f'(t) = 4 sqrt{3} cdot frac{ - frac{t}{2} + 6 }{ sqrt{t} (t + 12)^2 } ).Set ( f'(t) = 0 ):The numerator must be zero:( - frac{t}{2} + 6 = 0 ).Solve for ( t ):( - frac{t}{2} + 6 = 0 implies frac{t}{2} = 6 implies t = 12 ).So, critical point at ( t = 12 ).Now, check if this is a maximum.Compute the second derivative or test intervals around ( t = 12 ).Let me test values around ( t = 12 ):For ( t < 12 ), say ( t = 10 ):Numerator: ( -5 + 6 = 1 > 0 ), so ( f'(t) > 0 ).For ( t > 12 ), say ( t = 14 ):Numerator: ( -7 + 6 = -1 < 0 ), so ( f'(t) < 0 ).Therefore, ( t = 12 ) is a point of maximum.So, the maximum area occurs when ( t = 12 ).Recall that ( t = m^2 + 4 = 12 implies m^2 = 8 implies m = pm 2 sqrt{2} ).Now, compute the maximum area ( S ):( S = frac{4 sqrt{3} sqrt{12} }{12 + 12} = frac{4 sqrt{3} cdot 2 sqrt{3} }{24} = frac{4 cdot 2 cdot 3 }{24} = frac{24}{24} = 1 ).So, the maximum area is 1.Wait, that seems too clean. Let me verify.Wait, ( sqrt{12} = 2 sqrt{3} ), so:( S = frac{4 sqrt{3} cdot 2 sqrt{3} }{24} = frac{8 cdot 3 }{24} = frac{24}{24} = 1 ).Yes, that's correct.Therefore, the maximum area of triangle ( POQ ) is 1.**Final Answer**(1) The standard equation of ellipse ( C ) is (boxed{dfrac{x^2}{4} + y^2 = 1}).(2) The maximum area of triangle ( triangle POQ ) is (boxed{1}).

answer：To solve the expression (frac{81,!000^3}{27,!000^3}), I start by simplifying the fraction inside the exponent. Dividing both the numerator and the denominator by 1,000, I get (frac{81}{27}), which simplifies to 3. This reduces the original expression to (3^3). Calculating this, I find that (3^3 = 27). Therefore, the solution is 27.

answer：First, I observe that each term in the expression (3x(x-5) + 7(x-5) - 2(x-5)) contains the common factor ((x-5)). This suggests that I can factor out ((x-5)) from all the terms.After factoring out ((x-5)), the expression becomes ((3x + 7 - 2)(x-5)). Next, I simplify the coefficients inside the parentheses by combining like terms: (7 - 2 = 5).This results in the simplified expression ((3x + 5)(x-5)), which is the fully factored form of the original expression.

answer：Okay, so I've got this problem here where I'm given some logarithmic values and I need to compare them. Let me try to figure this out step by step.First, the problem states that (5^{7} = 78125) and (47^{3} = 103823). Then, it defines three logarithms: (a = log_{5}2), (b = log_{13}3), and (c = log_{47}5). I need to compare (a), (b), and (c) and determine their order from smallest to largest.Alright, so I know that logarithms can be tricky, but maybe I can convert them into exponential form to make things clearer. Let's see:1. (a = log_{5}2) implies that (5^{a} = 2).2. (b = log_{13}3) implies that (13^{b} = 3).3. (c = log_{47}5) implies that (47^{c} = 5).Hmm, so now I have these exponential equations. Maybe I can use the given values (5^{7} = 78125) and (47^{3} = 103823) to help me compare (a), (b), and (c).Let me start by comparing (b) and (c). So, (13^{b} = 3) and (47^{c} = 5). I wonder if I can relate these somehow. Maybe if I raise both sides of these equations to some power to make the bases similar or something.Wait, if I take (13^{b} = 3) and raise both sides to the power of 3, I get ((13^{b})^{3} = 3^{3}), which is (13^{3b} = 27). Similarly, if I take (47^{c} = 5) and raise both sides to the power of 2, I get ((47^{c})^{2} = 5^{2}), which is (47^{2c} = 25).Now, looking at (13^{3b} = 27) and (47^{2c} = 25), I can see that 27 is greater than 25. So, (13^{3b} > 47^{2c}). Hmm, but how does that help me compare (b) and (c)?Maybe I can take the logarithm of both sides to bring down the exponents. Let's try that. Taking the logarithm base 10 of both sides:(log(13^{3b}) > log(47^{2c}))Using the logarithm power rule, this becomes:(3b cdot log(13) > 2c cdot log(47))So, (3b > frac{2c cdot log(47)}{log(13)})Hmm, I need to figure out the approximate values of (log(13)) and (log(47)) to see if I can find a relationship between (b) and (c). I remember that (log(10) = 1), (log(100) = 2), so (log(13)) is a little more than 1.1, and (log(47)) is a little less than 1.7.Wait, let me check that. Actually, (log(10) = 1), (log(100) = 2), so (log(13)) is approximately 1.113 and (log(47)) is approximately 1.672.So plugging these in:(3b > frac{2c cdot 1.672}{1.113})Calculating the right side:(2 times 1.672 = 3.344)(3.344 / 1.113 ≈ 2.999), which is roughly 3.So, (3b > 3c), which simplifies to (b > c). Okay, so (b) is greater than (c). Got that.Now, let's compare (a) and (b). So, (a = log_{5}2) and (b = log_{13}3). I need to see which one is bigger.Again, maybe I can convert these into exponential form:(5^{a} = 2) and (13^{b} = 3).I wonder if I can relate these exponents somehow. Maybe if I express both sides with the same base or something.Alternatively, I can use the change of base formula to express these logarithms in terms of natural logarithms or base 10 logarithms.Let me try that. Using the change of base formula:(a = frac{log 2}{log 5}) and (b = frac{log 3}{log 13}).So, (a = frac{log 2}{log 5}) and (b = frac{log 3}{log 13}).I know that (log 2 ≈ 0.3010), (log 5 ≈ 0.6990), (log 3 ≈ 0.4771), and (log 13 ≈ 1.1133).So, plugging these in:(a ≈ frac{0.3010}{0.6990} ≈ 0.430)(b ≈ frac{0.4771}{1.1133} ≈ 0.428)Wait, so (a ≈ 0.430) and (b ≈ 0.428). So, (a) is slightly larger than (b). Interesting.But wait, let me double-check my calculations because sometimes approximations can be misleading.Calculating (a = log_{5}2):We know that (5^{1} = 5) and (5^{0} = 1). Since 2 is between 1 and 5, (a) must be between 0 and 1. Similarly, (b = log_{13}3) is also between 0 and 1.But to get a better estimate, maybe I can use the given values (5^{7} = 78125) and (47^{3} = 103823). How can these help?Well, (5^{7} = 78125). So, (5^{7} = 78125), which is a large number. Similarly, (47^{3} = 103823), which is also a large number.But how does that help me with (a), (b), and (c)?Wait, maybe I can use these to find approximate values for the logarithms.Let me think. Since (5^{7} = 78125), then (log_{5}78125 = 7). But I need (log_{5}2). Hmm, maybe I can relate 2 to 78125 somehow.Alternatively, maybe I can use the fact that (5^{a} = 2). If I take the logarithm of both sides, I get (a cdot log 5 = log 2), so (a = frac{log 2}{log 5}), which is what I did earlier.Similarly, for (b = frac{log 3}{log 13}).But perhaps I can use the given exponents to find a relationship.Wait, another approach: Maybe I can express all logarithms in terms of a common base, say base 10, and then compare their approximate values.So, let's compute (a), (b), and (c) numerically.First, (a = log_{5}2). Using the change of base formula:(a = frac{log 2}{log 5} ≈ frac{0.3010}{0.6990} ≈ 0.430)Next, (b = log_{13}3):(b = frac{log 3}{log 13} ≈ frac{0.4771}{1.1133} ≈ 0.428)Lastly, (c = log_{47}5):(c = frac{log 5}{log 47} ≈ frac{0.6990}{1.6721} ≈ 0.418)So, from these approximate values:(c ≈ 0.418), (b ≈ 0.428), (a ≈ 0.430)Therefore, (c < b < a). So, the order is (c < b < a), which corresponds to option A.Wait, but earlier when I compared (b) and (c) using the exponential forms, I concluded that (b > c), which aligns with this numerical result. And then comparing (a) and (b), I saw that (a) is slightly larger than (b), which also aligns.So, putting it all together, the correct order is (c < b < a), which is option A.But just to make sure I didn't make any mistakes in my calculations, let me verify the approximate values again.Calculating (a = log_{5}2):We know that (5^{0.430} ≈ 2). Let's check:(5^{0.430} = e^{0.430 cdot ln 5} ≈ e^{0.430 cdot 1.6094} ≈ e^{0.692} ≈ 2.000). Yes, that checks out.Similarly, (b = log_{13}3 ≈ 0.428):(13^{0.428} = e^{0.428 cdot ln 13} ≈ e^{0.428 cdot 2.5649} ≈ e^{1.099} ≈ 3.000). That also checks out.And (c = log_{47}5 ≈ 0.418):(47^{0.418} = e^{0.418 cdot ln 47} ≈ e^{0.418 cdot 3.8501} ≈ e^{1.614} ≈ 5.025). Hmm, that's a bit off, but close enough considering the approximations.So, overall, my calculations seem consistent. Therefore, I'm confident that the correct answer is option A: (c < b < a).

answer：First, I need to simplify the complex number ( z = 1 + 2i + i^3 ).I know that ( i^3 = -i ), so substituting that in, the expression becomes:[z = 1 + 2i - i]Combining like terms, this simplifies to:[z = 1 + i]Now, to find the magnitude ( |z| ), I use the formula for the magnitude of a complex number ( a + bi ):[|z| = sqrt{a^2 + b^2}]Substituting ( a = 1 ) and ( b = 1 ):[|z| = sqrt{1^2 + 1^2} = sqrt{1 + 1} = sqrt{2}]Therefore, the magnitude of ( z ) is ( sqrt{2} ), which corresponds to choice C.

answer：Okay, so I have this problem here about Avery opening a flower shop. She ties 8 bunches of flowers, and each bunch has 9 flowers. The question is asking how many bunches she would have if she put 12 flowers in each bunch instead. Hmm, okay, let's break this down step by step.First, I need to figure out the total number of flowers Avery has. She has 8 bunches, and each bunch has 9 flowers. So, to find the total number of flowers, I should multiply the number of bunches by the number of flowers per bunch. That would be 8 times 9. Let me write that down:8 bunches × 9 flowers per bunch = 72 flowersOkay, so Avery has a total of 72 flowers. Now, the problem is asking how many bunches she would have if she put 12 flowers in each bunch instead of 9. So, I need to find out how many bunches of 12 flowers she can make from her total of 72 flowers.To do that, I should divide the total number of flowers by the new number of flowers per bunch. That would be 72 flowers divided by 12 flowers per bunch. Let me write that down:72 flowers ÷ 12 flowers per bunch = 6 bunchesWait, so if Avery puts 12 flowers in each bunch, she would have 6 bunches instead of 8. That makes sense because each bunch is now bigger, so she needs fewer bunches to hold all her flowers.Let me double-check my calculations to make sure I didn't make any mistakes. Starting with 8 bunches of 9 flowers each:8 × 9 = 72Yes, that's correct. Now, dividing the total number of flowers by the new bunch size:72 ÷ 12 = 6That also checks out. So, it seems like my answer is correct. Avery would have 6 bunches if she puts 12 flowers in each bunch.I think I've got it. The key steps were finding the total number of flowers and then dividing by the new bunch size to find out how many bunches she would have. It's all about understanding how changing the number of flowers per bunch affects the total number of bunches needed.