💡First, I observe the expression ( mn - mn^3 ) and notice that both terms have a common factor of ( mn ).I factor out ( mn ) from the expression, which gives me ( mn(1 - n^2) ).Next, I recognize that ( 1 - n^2 ) is a difference of squares, which can be factored further into ( (1 + n)(1 - n) ).Finally, I combine these factors to obtain the fully factorized form of the original expression.

💡Okay, so I need to figure out how much money Mary made last week from her three jobs. She does home cleaning, babysitting, and pet care. Let me break this down step by step.First, let's look at her earnings from cleaning homes. She earns 46 for each home she cleans. Last week, she cleaned 4 homes. So, to find out how much she made from cleaning, I need to multiply the amount she earns per home by the number of homes she cleaned. That would be 46 multiplied by 4. Let me do that calculation: 46 times 4 equals... hmm, 40 times 4 is 160, and 6 times 4 is 24, so 160 plus 24 is 184. So, she made 184 from cleaning homes.Next, let's calculate her earnings from babysitting. She earns 35 per day for babysitting, and she worked for 5 days. So, I need to multiply 35 by 5. Let me see, 35 times 5 is... 30 times 5 is 150, and 5 times 5 is 25, so 150 plus 25 is 175. Therefore, she made 175 from babysitting.Now, onto her pet care earnings. She gets 60 per day for taking care of pets, and she worked for 3 days. So, I need to multiply 60 by 3. That's straightforward: 60 times 3 is 180. So, she made 180 from pet care.To find the total amount she made last week, I need to add up all her earnings from the three jobs. That means adding 184 from cleaning, 175 from babysitting, and 180 from pet care. Let me add them one by one. First, 184 plus 175. Let's see, 184 plus 100 is 284, plus 75 is 359. Then, adding the last amount: 359 plus 180. 359 plus 100 is 459, plus 80 is 539. So, altogether, she made 539 last week.Wait, let me double-check my calculations to make sure I didn't make any mistakes. For cleaning: 46 times 4 is indeed 184. For babysitting: 35 times 5 is 175. For pet care: 60 times 3 is 180. Adding them up: 184 plus 175 is 359, and 359 plus 180 is 539. Yep, that seems correct.I think I've covered all the steps and made sure each calculation is accurate. So, Mary made a total of 539 last week from her three jobs.

💡First, I need to determine the time Jack spends mopping each room individually by dividing the area of each room by his mopping speed for that room.For the bathroom:- Area: 24 square feet- Mopping speed: 8 square feet per minute- Time = 24 / 8 = 3 minutesFor the kitchen:- Area: 80 square feet- Mopping speed: 10 square feet per minute- Time = 80 / 10 = 8 minutesFor the living room:- Area: 120 square feet- Mopping speed: 7 square feet per minute- Time = 120 / 7 ≈ 17.14 minutesFinally, I add up the time spent in each room to find the total mopping time:3 minutes (bathroom) + 8 minutes (kitchen) + 17.14 minutes (living room) = 28.14 minutes

💡Okay, so I need to find the complex conjugate of the complex number z, which is given as (-3 + i) divided by (2 + i). Hmm, complex conjugates... I remember that for any complex number a + bi, its conjugate is a - bi. So, if I can express z in the standard form a + bi, then the conjugate would just be a - bi. But wait, z is given as a fraction, so I can't directly apply the conjugate rule until I simplify it. Maybe I should first simplify z by performing the division. I think the way to do that is by multiplying both the numerator and the denominator by the conjugate of the denominator. That should help eliminate the imaginary part in the denominator.So, the denominator is 2 + i, and its conjugate is 2 - i. If I multiply both the top and bottom by 2 - i, that should work. Let me write that down:z = (-3 + i)/(2 + i) * (2 - i)/(2 - i)Okay, now I need to multiply out the numerators and the denominators. Let's start with the numerator:(-3 + i)(2 - i). I'll use the distributive property (FOIL) for this.First, multiply -3 by 2: that's -6.Then, -3 multiplied by -i: that's +3i.Next, i multiplied by 2: that's +2i.Finally, i multiplied by -i: that's -i².So, putting it all together: -6 + 3i + 2i - i².I remember that i² is equal to -1, so -i² is +1. So, substituting that in:-6 + 3i + 2i + 1.Combine like terms: (-6 + 1) + (3i + 2i) = -5 + 5i.Okay, so the numerator simplifies to -5 + 5i.Now, let's work on the denominator:(2 + i)(2 - i). Again, using the distributive property.First, 2 multiplied by 2: that's 4.Then, 2 multiplied by -i: that's -2i.Next, i multiplied by 2: that's +2i.Finally, i multiplied by -i: that's -i².So, putting it all together: 4 - 2i + 2i - i².Again, i² is -1, so -i² is +1. Substituting that in:4 - 2i + 2i + 1.Combine like terms: (4 + 1) + (-2i + 2i) = 5 + 0i = 5.So, the denominator simplifies to 5.Now, putting it all together, z = (-5 + 5i)/5.I can factor out a 5 in the numerator:z = 5(-1 + i)/5.The 5 in the numerator and denominator cancels out, leaving:z = -1 + i.Alright, so z simplifies to -1 + i. Now, to find the complex conjugate of z, I just need to change the sign of the imaginary part. The original z is -1 + i, so the conjugate should be -1 - i.Let me double-check my steps to make sure I didn't make any mistakes. I multiplied both numerator and denominator by the conjugate of the denominator, which is correct. Then I expanded both products, remembering that i² = -1. The numerator simplified to -5 + 5i, and the denominator to 5. Dividing both by 5 gave me -1 + i. Taking the conjugate by changing the sign of the imaginary part gives -1 - i. That seems right.I think I'm confident with this answer. The complex conjugate of z is -1 - i.

💡First, I recognize that each of the three integers leaves a specific remainder when divided by 30. Let's denote these integers as ( a ), ( b ), and ( c ).Given:[a equiv 14 pmod{30}, quad b equiv 11 pmod{30}, quad c equiv 19 pmod{30}]To find the remainder when the sum of these integers is divided by 30, I'll add the remainders:[14 + 11 + 19 = 44]Next, I'll determine the remainder of 44 when divided by 30:[44 div 30 = 1 text{ with a remainder of } 14]Therefore, the remainder when the sum of the three integers is divided by 30 is 14.

💡Okay, so I have this problem here about a rectangle with an area of 12 square meters, and the sides are whole numbers. I need to find the greatest possible perimeter. Hmm, let me think about how to approach this.First, I remember that the area of a rectangle is calculated by multiplying its length and width. So, if the area is 12, then the length times the width equals 12. Since both the length and width are whole numbers, I need to find pairs of whole numbers that multiply to 12.Let me list out the factors of 12. Starting from 1, because 1 times 12 is 12. Then, 2 times 6 is 12, and 3 times 4 is 12. I think that's all the pairs because after that, it would start repeating in reverse order, like 4 times 3, which is the same as 3 times 4.So, the possible pairs for length and width are (1,12), (2,6), and (3,4). Now, I need to find the perimeter for each of these pairs because the question is asking for the greatest possible perimeter.I recall that the perimeter of a rectangle is calculated by adding up all the sides, which is 2 times the length plus 2 times the width. So, the formula is P = 2*(length + width).Let me calculate the perimeter for each pair:1. For the pair (1,12): P = 2*(1 + 12) = 2*13 = 26 meters.2. For the pair (2,6): P = 2*(2 + 6) = 2*8 = 16 meters.3. For the pair (3,4): P = 2*(3 + 4) = 2*7 = 14 meters.Now, comparing these perimeters: 26, 16, and 14. The largest one is 26 meters. So, the greatest possible perimeter is 26 meters.Wait, let me double-check to make sure I didn't miss any other factor pairs. 12 is a relatively small number, so the factors shouldn't be too many. I think I covered all the possible whole number pairs that multiply to 12. Yeah, I think that's correct.So, yeah, the answer should be 26 meters. That seems right.