Two matrices can only be multiplied if the second matrix has the same number of rows than the first one of columns. So if: A is a matrix M×N. B is a matrix M×K. We can't multiply A*B, because B does not have the same number of rows as A has of columns. if instead, B was N×K, the multiplication would be possible.Mar 26, 2017 · Step by step solution : Step 1 : Skip Ad Equation at the end of step 1 : (8 • (x24)) - 33y6 Step 2 : Equation at the end of step 2 : 23x24 - 33y6 Step 3 : Trying to factor as a Difference of Squares : 3.1 Factoring: 8x24-27y6 Theory : A difference of two perfect squares, A2 - B2 can be factored into (A+B) • (A-B) Feb 19, 2022 · The correct answer is option (a) mc020-2.jpg. Step-by-step explanation: Given: mc020-1.jpg. To Find: Equivalent Expression. Step 1. Let, mc020-1.jpg means . Step 2. Simplifying the equation as. Step 3. Equating the above equation as follows, Then is equivalent to . Therefore, the correct answer is option (a) mc020-2.jpg. #SPJ2 Nov 6, 2017 · Click here 👆 to get an answer to your question ️ What is the value of mc023-1.jpg? 73 90 290 365 the pic is the .jpg mc011-1.jpg. a. What is the sum? mc006-1.jpg. c. Which expression is equivalent to the following complex fraction? mc020-1.jpg. d. What is the difference? mc002-1.jpg. d. What is the sum? mc004-1.jpg. c. What is the sum? mc003-1.jpg. b. What is the sum? mc009-1.jpg. a. Which expression is equivalent to mc006-1.jpg? a. What is the product? mc001 ...1. 3-3 XRD slides for note taking.pdf. 14. Tooth Whitening Its Effects and Why It Causes Of Sensitivity.docx. 4. Practical Application of Skills Assignment 5.docx. 3. Utilitarian Theories A utilitarian theory of ethics evaluates the rightness or. 27.richard bought 3 slices of cheese pizza and 2 sodas for $8.75. Jordan bought 2 slices of cheese pizza and 4 sodas for $8.50. How much would an order of 1 slice of cheese pizza and 3 sodas cost? A. $3.25 B. $5.25 C. $7.75 D. $7.25Study with Quizlet and memorize flashcards containing terms like Which expression is equivalent to 9-2?, What is the quotient of mc009-1.jpg?, Which expression is equivalent to mc024-1.jpg? and more. Fresh features from the #1 AI-enhanced learning platform.You'll get a detailed solution from a subject matter expert that helps you learn core concepts. Question: 1) Identify the reagent (s) needed to complete the following reaction scheme. NaOH, Cl2 HOAc, Cl2. 1) Identify the reagent (s) needed to complete the following reaction scheme. 2) What is the most likely product of the following reaction?1. 3-3 XRD slides for note taking.pdf. 14. Tooth Whitening Its Effects and Why It Causes Of Sensitivity.docx. 4. Practical Application of Skills Assignment 5.docx. 3. Utilitarian Theories A utilitarian theory of ethics evaluates the rightness or. 27.Dec 28, 2016 · richard bought 3 slices of cheese pizza and 2 sodas for $8.75. Jordan bought 2 slices of cheese pizza and 4 sodas for $8.50. How much would an order of 1 slice of cheese pizza and 3 sodas cost? A. $3.25 B. $5.25 C. $7.75 D. $7.25 B. The equation mc024-1.jpg shows the relationship between a planet’s orbital period, T, and the planet’s mean distance from the sun, A, in astronomical units, AU. If planet Y is twice the mean distance from the sun as planet X, by what factor is the orbital period increased? D. Which of the following is equivalent to mc003-1.jpg? D.What is the product? mc001-1.jpg rotate abelabarca03 is waiting for your help. Add your answer and earn points. question 2 people found it helpful Amrit111Raj82 In mathematics, a product is the result of multiplication, or an expression that identifies factors to be multiplied.Mar 29, 2019 · The two exact solutions of mc017-1.jpg for mc017-2.jpg are: mc017-10.jpg and mc017-11.jpg. To solve the equation mc017-1.jpg for mc017-2.jpg, we first need to simplify the expression under the square root. We have:-3.jpg. To solve for x, we can square both sides of the equation: mc017-5.jpg Expanding the terms on the left-hand side and ... To rationalize the denominator of mc005-1.jpg , you should multiply the expression by which fraction? C. What is the following quotient? sqrt 6/sqrt 5 + sqrt 11/sqrt 3. C. What is the following quotient? 2-sqrt 8/ 4 + sqrt 12. C. Multiplying mc006-1.jpg by which fraction will produce an equivalent fraction with a rational denominator? B. What ... Jun 21, 2021 · 21.06.2021 Math Secondary School verified answered • expert verified What is the product? mc018-1.jpg mc018-2.jpg mc018-3.jpg mc018-4.jpg mc018-5 some of them are the answers on the bottom on top is the question Expert-Verified Answer question 1 person found it helpful sindhujammu1234 Step-by-step explanation: Where can you find your state-specific Lottery information to sell Lottery tickets and redeem winning Lottery tickets? (Select all that apply.) 1. Barcode and Quick Reference Guide 2. Lottery Terminal Handbook 3. Lottery vending machine 4. OneWalmart using Handheld/BYODJun 11, 2021 · What is the product? mc002-1.jpg See answers Advertisement Advertisement dobesc10088 dobesc10088 ... IF a+1/a=2 , find a to the power 4+1/a to the power 4 Click here 👆 to get an answer to your question ️ If f(x) = x2 – 1 and g(x) = 2x – 3, what is the domain of mc015-1.jpg?es024-1.jpg A similar right triangle would be created by a run of 4 and a rise of, The graph shows a line and two similar triangles. mc018-1.jpg What is the equation of the line? and more. Study with Quizlet and memorize flashcards containing terms like In the figure below, triangle GHI is similar to triangle MHL.Jan 10, 2022 · Which pair shows equivalent expressions? mc019-1. Jpg mc019-2. Jpg mc019-3. Jpg mc019-4. Jpg. Get the answers you need, now! Check : m 1 is not a square !! Ruling : Binomial can not be factored as the difference of two perfect squares. Trying to factor as a Difference of Cubes: 1.2 Factoring: mc 18-jpg Theory : A difference of two perfect cubes, a 3 - b 3 can be factored into (a-b) • (a 2 +ab +b 2) Proof : (a-b)•(a 2 +ab+b 2) = a 3 + a 2 b + ab 2-ba 2-b 2 a-b 3 =Dec 21, 2018 · The polynomial equation mc018-1.jpg has complex roots mc018-2.jpg. What is the other root? Use a graphing calculator and a system of equations. - 7228287 Answer: 3+6i. Step-by-step explanation: To divide we multiply by the conjugate of denominator. conjugate of 2+i is 2-i. Multiply top and bottom by 2-iStudy with Quizlet and memorize flashcards containing terms like Which expression is equivalent to 9-2?, What is the quotient of mc009-1.jpg?, Which expression is equivalent to mc024-1.jpg? and more. Fresh features from the #1 AI-enhanced learning platform.56 people found it helpful. jim1021. 13009-1.167 is the expression equivalent to mc009-1.jpg.Study with Quizlet and memorize flashcards containing terms like A stable atom that has a large nucleus most likely contains (Atoms and the Atomic Theory), Which statement best describes the light that will cause emission of electrons from a given metal through the photoelectric effect? (Atoms and the Atomic Theory), The atomic symbol mc025-1.jpg represents potassium-40 (K-40), a radioactive ...Which of the following classifies the sequence? B. What is the next term in the sequence below? -324. A. Which formula can be used to find the nth term of a geometric sequence where the fifth term is mc018-1.jpg and the common ratio is mc018-2.jpg? A. What is the common ratio of the sequence below? 2/3, 1/6. Medicines are resistant to antibiotics., The pedigree chart below shows the inheritance of a trait in a family of cats. mc018-1.jpg Which pair of individuals represents a mating that will result in inbreeding? individuals 1 and 6 individuals 3 and 4 individuals 3 and 6 individuals 6 and 14, Frank and Maria both had normal vision.Study with Quizlet and memorize flashcards containing terms like Select the one TRUE statement below regarding the melting temperature (Tm) of DNA., Two amphipathic α helices in a dilute aqueous solution are likely to:, If a Glu sidechain is in a hydrophobic environment, would you expect that the pKa of this sidechain would be higher or lower than the pKa for the sidechain in aqueous solution ...Study with Quizlet and memorize flashcards containing terms like Which is a coefficient matrix for the system of linear equations? mc014-1.jpg mc014-2.jpg mc014-3.jpg mc014-4.jpg mc014-5.jpg, What is the determinant of the coefficient matrix of the system ? -158 -117 65 117, Suppose matrix D is the inverse of matrix C. Which of the following is equal to D? CDD CCD DDD CCC and more.The difference between the polynomials, is expressed as: . Given the polynomials:. We are required to find their difference which can be solved as follows:. First, open the bracket by multiplying every term you have in by -1:what is mc001-2.jpg. A. If es002-1.jpgand g (x) = 1 - x2, then what. do you notice about the domain of es002-2.jpg ? The domain of f, and thus the range of g, is restricted to values greater than or equal to 3. If 1 minus x squared is greater than or equal to 3, then x squared must be less than -2.Unlock the answer. question. The amount of a sample remaining after t days is given by the equation mc004-1.jpg, where A is the initial amount of the sample and h is the half-life, in days, of the substance. A sample contains 60% of its original amount of Fermium-257. The half-life of Fermium-257 is about 100 days.what is mc001-2.jpg. A. If es002-1.jpgand g (x) = 1 - x2, then what. do you notice about the domain of es002-2.jpg ? The domain of f, and thus the range of g, is restricted to values greater than or equal to 3. If 1 minus x squared is greater than or equal to 3, then x squared must be less than -2. Identify the reference angle es001-1. Jpg for each given angle, es001-2. Jpg. Es001-3. Jpg degrees. Es002-1. Jpg degrees. Es003-1. Jpg degrees.Use a graphing calculator and a system of equations. -2, 1, 4. The polynomial equation mc020-1.jpg has complex roots mc020-2.jpg. What are the other roots? Use a graphing calculator and a system of equations. -2, 2. The graph of this system of equations is used to solve mc006-1.jpg. x coordinates of the intersection points.(1): "c0" was replaced by "c^0". (2): Dot was discarded near "1.j". Step 1 : Trying to factor as a Difference of Squares : 1.1 Factoring: mc 20-jpg Theory : A difference of two perfect squares, A 2 - B 2 can be factored into (A+B) • (A-B) Proof : (A+B) • (A-B) = A 2 - AB + BA - B 2 = A 2 - AB + AB - B 2 = A 2 - B 2B. The equation mc024-1.jpg shows the relationship between a planet’s orbital period, T, and the planet’s mean distance from the sun, A, in astronomical units, AU. If planet Y is twice the mean distance from the sun as planet X, by what factor is the orbital period increased? D. Which of the following is equivalent to mc003-1.jpg? D.Apr 23, 2021 · What is the product? mc001-1.jpg rotate abelabarca03 is waiting for your help. Add your answer and earn points. question 2 people found it helpful Amrit111Raj82 In mathematics, a product is the result of multiplication, or an expression that identifies factors to be multiplied. The lattice energy is -734 kJ/mol, the enthalpy of hydration for mc031-5.jpg is -499 kJ/mol, and the enthalpy of hydration for mc031-6.jpg is -284 kJ/mol. Use mc031-7.jpgHsol = -mc031-8.jpgHlat + mc031-9.jpgHhydr. The mc018-1.jpgHsol for NaF is 82 kJ/mol, and the lattice energy is -923 kJ/mol. If the hydration enthalpy of sodium is 21 kJ/mol ... Apr 23, 2021 · What is the product? mc001-1.jpg rotate abelabarca03 is waiting for your help. Add your answer and earn points. question 2 people found it helpful Amrit111Raj82 In mathematics, a product is the result of multiplication, or an expression that identifies factors to be multiplied. richard bought 3 slices of cheese pizza and 2 sodas for $8.75. Jordan bought 2 slices of cheese pizza and 4 sodas for $8.50. How much would an order of 1 slice of cheese pizza and 3 sodas cost? A. $3.25 B. $5.25 C. $7.75 D. $7.25question. To find an inverse matrix you first have to find the determinant and then put the absolute value of it under a 1. That's what is sitting outside in front of all those matrices. What you have to do now is multiply that determinant by what I teach to my students as the "mixed up matrix". They always remember from that.Medicines are resistant to antibiotics., The pedigree chart below shows the inheritance of a trait in a family of cats. mc018-1.jpg Which pair of individuals represents a mating that will result in inbreeding? individuals 1 and 6 individuals 3 and 4 individuals 3 and 6 individuals 6 and 14, Frank and Maria both had normal vision.Nov 6, 2017 · Click here 👆 to get an answer to your question ️ What is the value of mc023-1.jpg? 73 90 290 365 the pic is the .jpg Study with Quizlet and memorize flashcards containing terms like Consider the following chemical reaction: 2CH4(g) es002-1.jpg C2H2(g) + 3H2(g) Describe what is happening within the system when it is at equilibrium in terms of concentrations, reactions that occur, and reaction rates., Identify the correct equilibrium constant expression for this equation: 2NO(g) + O2(g) <-> 2NO2(g), Identify ...Where can you find your state-specific Lottery information to sell Lottery tickets and redeem winning Lottery tickets? (Select all that apply.) 1. Barcode and Quick Reference Guide 2. Lottery Terminal Handbook 3. Lottery vending machine 4. OneWalmart using Handheld/BYODDec 30, 2017 · richard bought 3 slices of cheese pizza and 2 sodas for $8.75. Jordan bought 2 slices of cheese pizza and 4 sodas for $8.50. How much would an order of 1 slice of cheese pizza and 3 sodas cost? A. $3.25 B. $5.25 C. $7.75 D. $7.25 the solution to the algebraic expression will be: sqrt (1-3x)=x+3. squaring both sides we get: 1-3x= (x+3)^2. 1-3x=x^2+6x+9. this can be written in quadratic form as: x^2+6x+3x+9-1=0. x^2+9x+8=0. factorizing the above we get:Study with Quizlet and memorize flashcards containing terms like Select the one TRUE statement below regarding the melting temperature (Tm) of DNA., Two amphipathic α helices in a dilute aqueous solution are likely to:, If a Glu sidechain is in a hydrophobic environment, would you expect that the pKa of this sidechain would be higher or lower than the pKa for the sidechain in aqueous solution ...Feb 5, 2021 · Which pair shows equivalent expressions? mc017-1.jpg mc017-2.jpg mc017-3.jpg mc017-4.jpg Terms in this set (50) Use completing the square to solve for mc024-1.jpg in the equation mc024-2.jpg. c x = 1 + 89. Solve for mc022-1.jpg in the equation mc022-2.jpg. d x= 1 + 47. Solve for mc018-1.jpg in the equation mc018-2.jpg. a x = 2 + 42. Which value must be added to the expression x2 + 16x to make it a perfect-square trinomial? 64.The two exact solutions of mc017-1.jpg for mc017-2.jpg are: mc017-10.jpg and mc017-11.jpg. To solve the equation mc017-1.jpg for mc017-2.jpg, we first need to simplify the expression under the square root. We have:-3.jpg. To solve for x, we can square both sides of the equation: mc017-5.jpg Expanding the terms on the left-hand side and ...Dec 21, 2018 · The polynomial equation mc018-1.jpg has complex roots mc018-2.jpg. What is the other root? Use a graphing calculator and a system of equations. - 7228287 Step-by-step explanation: Given: Polynomial. We have to find the product of the given polynomial. Consider the given polynomial. Apply distributive rule, Multiply with each term in brackets, we have, Apply exponent rule, Simplify, we have, Thus, The product of the given polynomial is.Identify the reference angle es001-1. Jpg for each given angle, es001-2. Jpg. Es001-3. Jpg degrees. Es002-1. Jpg degrees. Es003-1. Jpg degrees.Click here 👆 to get an answer to your question ️ If f(x) = x2 – 1 and g(x) = 2x – 3, what is the domain of mc015-1.jpg?Algebra. Equation Solver. Step 1: Enter the Equation you want to solve into the editor. The equation calculator allows you to take a simple or complex equation and solve by best method possible. Step 2: Click the blue arrow to submit and see the result!Moises is determining the solution to the system of equations that is show below. Equation representing line A: mc018-1.jpg Points on line B: (3, –12) and (9, –24) Step 1: Determine the slope for B: mc018-2.jpg Step 2: Determine the y-intercept for B: mc018-3.jpg Step 3: Write the equation in slope-intercept form: mc018-4.jpgDec 21, 2018 · The polynomial equation mc018-1.jpg has complex roots mc018-2.jpg. What is the other root? Use a graphing calculator and a system of equations. - 7228287 Check : m 1 is not a square !! Ruling : Binomial can not be factored as the difference of two perfect squares. Trying to factor as a Difference of Cubes: 1.2 Factoring: mc 18-jpg Theory : A difference of two perfect cubes, a 3 - b 3 can be factored into (a-b) • (a 2 +ab +b 2) Proof : (a-b)•(a 2 +ab+b 2) = a 3 + a 2 b + ab 2-ba 2-b 2 a-b 3 =Jan 12, 2018 · Click here 👆 to get an answer to your question ️ If f(x) = x2 – 1 and g(x) = 2x – 3, what is the domain of mc015-1.jpg? Study with Quizlet and memorize flashcards containing terms like What is the length of the hypotenuse in the triangle below? mc017-1.jpg, If a vertical line is dropped from the x-axis to the point (12, -9) in the diagram below, what is the value of sec mc022-1.jpg? mc022-2.jpg, Which of the following is cot(mc013-1.jpg) sec(mc013-2.jpg) in simplified form? and more.Which graph represents the function mc018-1.jpg? d. ... What is the product? mc001-1.jpg. a. Which expression is equivalent to mc005-1.jpgdivided by x + 1? a. The roots of the equation will be 2, 3 and 4. Remainder theorem of polynomial. Given the given polynomial function g(x) = x³- 9x² + 26 x -24. if the one root is equal to 2, then we can determine the other roots by synthetic division:Its small ziti dish uses 1 cup of sauce and mc022-1.jpg cups of cheese. Its large ziti dish uses 2 cups of sauce and 3 cups of cheese. The business has 100 cups of sauce and 400 cups of cheese on hand. It makes $6 profit on their small dishes and $5 profit on their large dishes. It wants to maximize the profit from selling the two sizes of ziti.Which pair shows equivalent expressions? mc017-1.jpg mc017-2.jpg mc017-3.jpg mc017-4.jpgCatabolic reactions are reactions that break larger molecules down into smaller molecules, and often result in the release of energy as a by product of the bonds being broken. In fact, metabolism (the breakdown of food) is a catabolic reaction. So the best answer would be the Answer no.4: A => B + C + EnergyConsider a triangle ABC like the one below. Suppose that a=34, b=53, and c=74. The figure is not drawn to scale.) Solve the triangle. Carry your intermediate computations to at least four decimal places, and round your answers to the nearest tenth.The difference between the polynomials, is expressed as: . Given the polynomials:. We are required to find their difference which can be solved as follows:. First, open the bracket by multiplying every term you have in by -1:Answer: 3+6i. Step-by-step explanation: To divide we multiply by the conjugate of denominator. conjugate of 2+i is 2-i. Multiply top and bottom by 2-iMar 15, 2022 · Answer 44 people found it helpful brailynnem0155 Answer: first one b, second c Explanation: This is not a multiple choice question It's C A cereal box manufacturer changes the size of the box to increase the amount of cereal it contains. The equations 12 + 7.6n and 6 + 8n, where n is the number of smaller boxes, are both representative of the amount of cereal that the new larger box contains. Like terms are the terms those have same variables. In the given polynomial, we have first term mn and second term -11mn. Both of the terms has mn variables with them. So, we can subtract mn - 11mn. mn could be written as 1mn. Therefore, 1mn -11mn = -10mn.Question: At 20°C, the ion-product constant of water, Kw, is mc018-1.jpg. What is the pH of pure water at 20°C? What is the pH of pure water at 20°C? At 20°C, the ion-product constant of water, Kw, is mc018-1.jpg.Moises is determining the solution to the system of equations that is show below. Equation representing line A: mc018-1.jpg Points on line B: (3, –12) and (9, –24) Step 1: Determine the slope for B: mc018-2.jpg Step 2: Determine the y-intercept for B: mc018-3.jpg Step 3: Write the equation in slope-intercept form: mc018-4.jpgStudy with Quizlet and memorize flashcards containing terms like Consider the following chemical reaction: 2CH4(g) es002-1.jpg C2H2(g) + 3H2(g) Describe what is happening within the system when it is at equilibrium in terms of concentrations, reactions that occur, and reaction rates., Identify the correct equilibrium constant expression for this equation: 2NO(g) + O2(g) <-> 2NO2(g), Identify ...
Apr 11, 2017 · Verified answer. Where can you find your state-specific Lottery information to sell Lottery tickets and redeem winning Lottery tickets? (Select all that apply.) 1. Barcode and Quick Reference Guide 2. Lottery Terminal Handbook 3. Lottery vending machine 4. OneWalmart using Handheld/BYOD. heart. . Voueur house.tv
Dimensional Analysis 1. I have 470 milligrams of table salt, which is the chemical compound NaCl. How many liters of NaCl solution can I make if I want the solution to be 0.90% NaCl? (9 grams of salt per 1000 grams of solution). The density of the NaCl solution is 1.0 g solution/mL solution.Verified answer. Where can you find your state-specific Lottery information to sell Lottery tickets and redeem winning Lottery tickets? (Select all that apply.) 1. Barcode and Quick Reference Guide 2. Lottery Terminal Handbook 3. Lottery vending machine 4. OneWalmart using Handheld/BYOD. heart.Give the worksheet consolidation entries relating to inventory and cost of goods sold needed to prep. 1. Peterson_s Master NCLEX PN.pdf. 478. 5. Identify hazards, assess and control risks and evaluate and maintain WHS.docx. 9. CITATION Ian22 l 1033 812 Benefits of ERP Systems The extensive feature sets of. 115.Dec 28, 2016 · richard bought 3 slices of cheese pizza and 2 sodas for $8.75. Jordan bought 2 slices of cheese pizza and 4 sodas for $8.50. How much would an order of 1 slice of cheese pizza and 3 sodas cost? A. $3.25 B. $5.25 C. $7.75 D. $7.25 Jan 10, 2022 · Which pair shows equivalent expressions? mc019-1. Jpg mc019-2. Jpg mc019-3. Jpg mc019-4. Jpg. Get the answers you need, now! 1.1 Factoring: mc 3-jpg Theory : A difference of two perfect cubes, a 3 - b 3 can be factored into (a-b) • (a 2 +ab +b 2) Proof : (a-b)•(a 2 +ab+b 2) = a 3 + a 2 b + ab 2-ba 2-b 2 a-b 3 = a 3 +(a 2 b-ba 2)+(ab 2-b 2 a)-b 3 = a 3 + 0 + 0-b 3 = a 3-b 3 Check : m 1 is not a cube !! Ruling : Binomial can not be factored as the difference of two ...richard bought 3 slices of cheese pizza and 2 sodas for $8.75. Jordan bought 2 slices of cheese pizza and 4 sodas for $8.50. How much would an order of 1 slice of cheese pizza and 3 sodas cost? A. $3.25 B. $5.25 C. $7.75 D. $7.25What is the second product in the reaction: a. fatty acid salt. b. ester. c. alcohol. d. aldehydeWhat is the product? mc001-1.jpg. A. Simplify the expression below. es027-1.jpg. What is the numerator of the simplified expression? N. ... mc018-1.jpg. C.Mar 29, 2019 · The two exact solutions of mc017-1.jpg for mc017-2.jpg are: mc017-10.jpg and mc017-11.jpg. To solve the equation mc017-1.jpg for mc017-2.jpg, we first need to simplify the expression under the square root. We have:-3.jpg. To solve for x, we can square both sides of the equation: mc017-5.jpg Expanding the terms on the left-hand side and ... Jun 7, 2018 · Click here 👆 to get an answer to your question ️ One factor of mc018-1.jpg is (x + 7). What are all the roots of the function? Use the Remainder Theorem. Answer to Predict the major product of the following reaction.Which graph represents the function mc018-1.jpg? d. ... What is the product? mc001-1.jpg. a. Which expression is equivalent to mc005-1.jpgdivided by x + 1? a. Aug 22, 2018 · 56 people found it helpful. jim1021. 13009-1.167 is the expression equivalent to mc009-1.jpg. Catabolic reactions are reactions that break larger molecules down into smaller molecules, and often result in the release of energy as a by product of the bonds being broken. In fact, metabolism (the breakdown of food) is a catabolic reaction. So the best answer would be the Answer no.4: A => B + C + EnergyFree math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor..
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