💡First, I need to subtract the complex numbers (3 - 5i) from (6 + 2i).I'll start by subtracting the real parts of the two complex numbers: (6 - 3 = 3).Next, I'll subtract the imaginary parts: (2i - (-5i) = 2i + 5i = 7i).Finally, I'll combine the results of the real and imaginary parts to get the final answer: (3 + 7i).

💡First, I will identify the initial number of each type of jewelry that Jessy has:- **Necklaces:** 10- **Earrings:** 15- **Bracelets:** 5- **Rings:** 8Next, I will calculate the purchases made at Store A:- **Necklaces:** 10 additional- **Earrings:** 2/3 of the initial 15, which is ( frac{2}{3} times 15 = 10 )- **Bracelets:** 3 additionalThen, I will determine the purchases made at Store B:- **Rings:** Twice the original amount, which is ( 2 times 8 = 16 )- **Necklaces:** 4 additionalAfter that, I will account for the gift from Jessy's mother:- **Earrings:** 1/5 of the number purchased at Store A, which is ( frac{1}{5} times 10 = 2 )Finally, I will sum up all the quantities for each type of jewelry:- **Necklaces:** ( 10 + 10 + 4 = 24 )- **Earrings:** ( 15 + 10 + 2 = 27 )- **Bracelets:** ( 5 + 3 = 8 )- **Rings:** ( 8 + 16 = 24 )Adding all these together gives the total number of jewelry pieces:( 24 + 27 + 8 + 24 = 83 )

💡First, I need to determine the total percentage of sales that were pens, pencils, and erasers. I add the percentages of pens, pencils, and erasers together:25% (pens) + 30% (pencils) + 20% (erasers) = 75%.Since the total percentage of all sales should add up to 100%, I subtract the combined percentage of pens, pencils, and erasers from 100% to find the percentage of other items:100% - 75% = 25%.Therefore, 25% of the sales were not pens, pencils, or erasers.

💡First, I need to determine the price the trader paid for the car after a 20% discount. If the original price is ( P ), then the purchase price is ( 0.80P ).Next, the trader sold the car at a 70% increase on the purchase price. This means the selling price is ( 0.80P + 0.70 times 0.80P = 1.36P ).To find the profit percentage based on the original price, I calculate the difference between the selling price and the original price, which is ( 1.36P - P = 0.36P ).Finally, I determine the profit percentage by dividing the profit by the original price and multiplying by 100: ( frac{0.36P}{P} times 100% = 36% ).

💡Okay, so I have this problem here: If ( n diamondsuit m = n^2 m^3 ), what is ( frac{5 diamondsuit 3}{3 diamondsuit 5} )? The answer choices are ( textbf{(A)} frac{1}{5} ), ( textbf{(B)} frac{2}{5} ), ( textbf{(C)} frac{3}{5} ), ( textbf{(D)} frac{4}{5} ), and ( textbf{(E)} 1 ). Hmm, let me try to figure this out step by step.First, I need to understand what the operation ( diamondsuit ) means. It's defined as ( n diamondsuit m = n^2 m^3 ). So, for any two numbers ( n ) and ( m ), the operation ( diamondsuit ) takes ( n ) squared and multiplies it by ( m ) cubed. Got it.Now, the problem is asking for the ratio of ( 5 diamondsuit 3 ) to ( 3 diamondsuit 5 ). That is, I need to compute ( frac{5 diamondsuit 3}{3 diamondsuit 5} ). Let me break this down.Let's start by calculating ( 5 diamondsuit 3 ). Using the definition of ( diamondsuit ), this should be ( 5^2 times 3^3 ). Calculating each part:- ( 5^2 = 25 )- ( 3^3 = 27 )So, ( 5 diamondsuit 3 = 25 times 27 ). Let me compute that: 25 times 27. Hmm, 25 times 20 is 500, and 25 times 7 is 175, so adding those together gives 500 + 175 = 675. So, ( 5 diamondsuit 3 = 675 ).Next, I need to calculate ( 3 diamondsuit 5 ). Again, using the definition, this is ( 3^2 times 5^3 ). Let's compute each part:- ( 3^2 = 9 )- ( 5^3 = 125 )So, ( 3 diamondsuit 5 = 9 times 125 ). Let me do that multiplication: 9 times 100 is 900, and 9 times 25 is 225, so adding those gives 900 + 225 = 1125. Therefore, ( 3 diamondsuit 5 = 1125 ).Now, I need to find the ratio of these two results: ( frac{675}{1125} ). Let me simplify this fraction. Both 675 and 1125 are divisible by 225, I think. Let me check:- 675 divided by 225 is 3.- 1125 divided by 225 is 5.So, ( frac{675}{1125} = frac{3}{5} ). Therefore, the ratio ( frac{5 diamondsuit 3}{3 diamondsuit 5} ) simplifies to ( frac{3}{5} ).Looking back at the answer choices, ( frac{3}{5} ) corresponds to option ( textbf{(C)} ). So, I think the correct answer is ( textbf{(C)} frac{3}{5} ).Wait, just to make sure I didn't make any calculation errors. Let me double-check my computations.For ( 5 diamondsuit 3 ):- ( 5^2 = 25 )- ( 3^3 = 27 )- ( 25 times 27 = 675 ) (Yes, that's correct.)For ( 3 diamondsuit 5 ):- ( 3^2 = 9 )- ( 5^3 = 125 )- ( 9 times 125 = 1125 ) (That's also correct.)Then, the ratio ( frac{675}{1125} ). Dividing numerator and denominator by 225:- ( 675 ÷ 225 = 3 )- ( 1125 ÷ 225 = 5 )- So, ( frac{3}{5} ) is indeed the simplified form.Alright, I feel confident that my answer is correct. It's always good to check, especially when dealing with exponents and multiplication, as it's easy to make a small mistake there.Another way to approach this problem might be to look for patterns or properties of the operation ( diamondsuit ) that could simplify the ratio without calculating each term separately. Let me see if that's possible.Given ( n diamondsuit m = n^2 m^3 ), then ( frac{n diamondsuit m}{m diamondsuit n} = frac{n^2 m^3}{m^2 n^3} ). Simplifying this, we can cancel out terms:- ( frac{n^2}{n^3} = frac{1}{n} )- ( frac{m^3}{m^2} = m )So, ( frac{n diamondsuit m}{m diamondsuit n} = frac{m}{n} ).Wait, that's interesting. So, in general, the ratio ( frac{n diamondsuit m}{m diamondsuit n} ) simplifies to ( frac{m}{n} ). Applying this to our specific case where ( n = 5 ) and ( m = 3 ), we get:( frac{5 diamondsuit 3}{3 diamondsuit 5} = frac{3}{5} ).That's the same result as before, but arrived at through a more general approach. This method might be quicker and less error-prone, especially if the numbers were larger or more complex.So, using this property, we can see that the ratio is simply ( frac{m}{n} ), which in this case is ( frac{3}{5} ). This confirms my earlier calculation.I think this is a solid approach. It's always nice when you can find a general formula or property that applies to a problem, as it can save time and reduce the chance of computational errors.In summary, whether I compute each term separately and then take the ratio or use the general property of the operation ( diamondsuit ), I arrive at the same answer: ( frac{3}{5} ). Therefore, I'm confident that the correct choice is ( textbf{(C)} frac{3}{5} ).**Final Answer**boxed{C}

💡Okay, so I have this problem where I need to find the length of the arc of the curve given by the equation ( y = -arccos(sqrt{x}) + sqrt{x - x^2} ) from ( x = 0 ) to ( x = frac{1}{4} ). Hmm, arc length. I remember that the formula for the arc length of a curve ( y = f(x) ) from ( x = a ) to ( x = b ) is:[L = int_{a}^{b} sqrt{1 + left( frac{dy}{dx} right)^2} , dx]So, first things first, I need to find the derivative of ( y ) with respect to ( x ). Let me write down the function again:[y = -arccos(sqrt{x}) + sqrt{x - x^2}]Alright, so I need to differentiate this function term by term. Let's start with the first term, ( -arccos(sqrt{x}) ). I remember that the derivative of ( arccos(u) ) with respect to ( x ) is ( -frac{u'}{sqrt{1 - u^2}} ). So, applying that here:Let ( u = sqrt{x} ), so ( u' = frac{1}{2sqrt{x}} ). Therefore, the derivative of ( -arccos(sqrt{x}) ) is:[-frac{d}{dx} arccos(sqrt{x}) = -left( -frac{1}{sqrt{1 - (sqrt{x})^2}} cdot frac{1}{2sqrt{x}} right) = frac{1}{2sqrt{x}sqrt{1 - x}}]Wait, hold on, I think I messed up the signs. The derivative of ( arccos(u) ) is ( -frac{u'}{sqrt{1 - u^2}} ), so since there's already a negative sign in front of ( arccos ), it should be:[-frac{d}{dx} arccos(sqrt{x}) = -left( -frac{1}{sqrt{1 - x}} cdot frac{1}{2sqrt{x}} right) = frac{1}{2sqrt{x}sqrt{1 - x}}]Yes, that seems right.Now, moving on to the second term, ( sqrt{x - x^2} ). Let me set ( v = x - x^2 ), so ( v' = 1 - 2x ). The derivative of ( sqrt{v} ) is ( frac{v'}{2sqrt{v}} ), so:[frac{d}{dx} sqrt{x - x^2} = frac{1 - 2x}{2sqrt{x - x^2}}]Okay, so putting it all together, the derivative ( frac{dy}{dx} ) is:[frac{dy}{dx} = frac{1}{2sqrt{x}sqrt{1 - x}} + frac{1 - 2x}{2sqrt{x - x^2}}]Hmm, let me see if I can simplify this expression. Notice that ( sqrt{x - x^2} = sqrt{x(1 - x)} ), so ( sqrt{x - x^2} = sqrt{x}sqrt{1 - x} ). Therefore, the second term becomes:[frac{1 - 2x}{2sqrt{x}sqrt{1 - x}}]So now, both terms have the same denominator, ( 2sqrt{x}sqrt{1 - x} ). Let me write them together:[frac{dy}{dx} = frac{1}{2sqrt{x}sqrt{1 - x}} + frac{1 - 2x}{2sqrt{x}sqrt{1 - x}} = frac{1 + (1 - 2x)}{2sqrt{x}sqrt{1 - x}} = frac{2 - 2x}{2sqrt{x}sqrt{1 - x}} = frac{1 - x}{sqrt{x}sqrt{1 - x}}]Simplifying further, ( frac{1 - x}{sqrt{x}sqrt{1 - x}} = sqrt{frac{1 - x}{x}} ). So, the derivative simplifies to:[frac{dy}{dx} = sqrt{frac{1 - x}{x}}]Alright, that seems manageable. Now, plugging this into the arc length formula:[L = int_{0}^{frac{1}{4}} sqrt{1 + left( sqrt{frac{1 - x}{x}} right)^2} , dx]Let me compute the expression inside the square root:[1 + left( sqrt{frac{1 - x}{x}} right)^2 = 1 + frac{1 - x}{x} = frac{x + 1 - x}{x} = frac{1}{x}]So, the integrand simplifies to:[sqrt{frac{1}{x}} = frac{1}{sqrt{x}}]Therefore, the arc length integral becomes:[L = int_{0}^{frac{1}{4}} frac{1}{sqrt{x}} , dx]This is a standard integral. The integral of ( frac{1}{sqrt{x}} ) is ( 2sqrt{x} ). So, evaluating from 0 to ( frac{1}{4} ):[L = left[ 2sqrt{x} right]_{0}^{frac{1}{4}} = 2sqrt{frac{1}{4}} - 2sqrt{0} = 2 cdot frac{1}{2} - 0 = 1]Wait, so the arc length is 1? That seems surprisingly simple. Let me double-check my steps.First, the derivative calculation: I started with ( y = -arccos(sqrt{x}) + sqrt{x - x^2} ). The derivative of ( -arccos(sqrt{x}) ) is ( frac{1}{2sqrt{x}sqrt{1 - x}} ), and the derivative of ( sqrt{x - x^2} ) is ( frac{1 - 2x}{2sqrt{x - x^2}} ). Combining these gives ( sqrt{frac{1 - x}{x}} ). That seems correct.Then, plugging into the arc length formula, I got ( sqrt{1 + left( sqrt{frac{1 - x}{x}} right)^2} = sqrt{frac{1}{x}} ). That also seems right because ( 1 + frac{1 - x}{x} = frac{x + 1 - x}{x} = frac{1}{x} ).So, integrating ( frac{1}{sqrt{x}} ) from 0 to ( frac{1}{4} ) gives 1. Hmm, okay, maybe it is correct. The function simplifies nicely, so the arc length ends up being 1.I guess I was expecting something more complicated, but sometimes things cancel out nicely. I don't see any mistakes in my calculations, so I think the arc length is indeed 1.