X times 1 - , the numerator is 3, and the denominator is 8. A more illustrative example could involve a pie with 8 slices. 1 of those 8 slices would constitute the numerator of a fraction, while the total of 8 slices that comprises the whole pie would be the denominator. If a person were to eat 3 slices, the remaining fraction of the pie would therefore be

 
Remember, 2x times 4x is the same thing as-- you can rearrange the order of multiplication. This is the same thing as 2 times 4, times x times x. Which is the same thing as 8 times x squared. Remember, x to the 1, times x to the 1, add the exponents. I mean, you know x times x is x squared. So this first term is going to be 8x squared.. Chat.open a i.com

Which expression is equivalent to log Subscript 12 Baseline StartFraction x Superscript 4 Baseline StartRoot x cubed minus 2 EndRoot Over (x + 1) Superscript 5 Baseline EndFraction? 4 log Subscript 12 Baseline x + one-half log Subscript 12 Baseline (x cubed minus 2) minus 5 log Subscript 12 Baseline (x times 1)Simplify 1/ ( square root of x) 1 √x 1 x. Multiply 1 √x 1 x by √x √x x x. 1 √x ⋅ √x √x 1 x ⋅ x x. Combine and simplify the denominator.In order to show that T is a function, you need to prove that for each pair of subsets A and B of X there is one and only one subset U of X\times X such that T(A,B)=U, that is, ((A,B),U) \in T ... An exponent is the number of times to multiply a number by itself. Write an exponent as a raised number. In the number 2 4 (2 to the exponent 4, or 2 to the power of 4), the ‘4’ is the exponent. The ‘2’ is the number to multiply by itself 4 times over. In this case 2 x 2 x 2 x 2 = 16.Sounds tough, but once you have mastered the 10× table, it is just a few steps away. Firstly, 11× is mostly easy: from 11×2 to 11×9 you just put the two digits together. 11×2=22, 11×3=33, ..., 11×9=99. And of course 2×, 5× and 10× just follow their simple rules you know already. So it just leaves these to remember:Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor.Simplify x^ (1/2)*x^ (1/2) x1 2 ⋅ x1 2 x 1 2 ⋅ x 1 2. Multiply x1 2 x 1 2 by x1 2 x 1 2 by adding the exponents. Tap for more steps... x1 x 1. Simplify x1 x 1.Understand Negative numbers, one step at a time. Step by steps for fractions, factoring, and prime factorization. Enter your math expression. x2 − 2x + 1 = 3x − 5. Get Chegg Math Solver. $9.95 per month (cancel anytime). See details.Solve your math problems using our free math solver with step-by-step solutions. Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more. Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor. Simplify 1/ ( square root of x) 1 √x 1 x. Multiply 1 √x 1 x by √x √x x x. 1 √x ⋅ √x √x 1 x ⋅ x x. Combine and simplify the denominator., the numerator is 3, and the denominator is 8. A more illustrative example could involve a pie with 8 slices. 1 of those 8 slices would constitute the numerator of a fraction, while the total of 8 slices that comprises the whole pie would be the denominator. If a person were to eat 3 slices, the remaining fraction of the pie would therefore beWe could have factored this numerator as x plus 4 times x plus 1. 4 times 1 is 4. 4 plus 1 is 5, all of that over x plus 4. That cancels out and you're left just with x plus 1. Either way would have worked, but the algebraic long division will always work, even if you can't cancel out factors like that, even if you did have a remainder.Which expression is equivalent to log subscript 12 baseline startfraction x superscript 4 baseline startroot x cubed minus 2 endroot over (x 1) superscript 5 baseline endfraction? 4 log subscript 12 baseline x one-half log subscript 12 baseline (x cubed minus 2) minus 5 log subscript 12 baseline (x times 1) 4 log subscript 12 baseline x one-half log subscript 12 baseline startfraction x cubed ...Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. For math, science, nutrition, history ...Algebra. Simplify 1/2x^ (-1/2) 1 2 x−1 2 1 2 x - 1 2. Rewrite the expression using the negative exponent rule b−n = 1 bn b - n = 1 b n. 1 2 ⋅ 1 x1 2 1 2 ⋅ 1 x 1 2. Combine. 1⋅1 2x1 2 1 ⋅ 1 2 x 1 2. Multiply 1 1 by 1 1. 1 2x1 2 1 2 x 1 2.Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor.Simplify x^ (1/2)*x^ (1/2) x1 2 ⋅ x1 2 x 1 2 ⋅ x 1 2. Multiply x1 2 x 1 2 by x1 2 x 1 2 by adding the exponents. Tap for more steps... x1 x 1. Simplify x1 x 1.Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.Free Algebraic Properties Calculator - Simplify radicals, exponents, logarithms, absolute values and complex numbers step-by-step X-Times is a technology company that creates high-end digital chip design solutions. The company is focusing on independent research and development of the digital implementation EDA platform in accordance with the 3S concept (Smart, Speedy, Simple), including a new generation of layout and wiring technology while providing high-end digital chipIn order to show that T is a function, you need to prove that for each pair of subsets A and B of X there is one and only one subset U of X\times X such that T(A,B)=U, that is, ((A,B),U) \in T ... What is the ''right" norm for the Banach space tensor product in this situation?Sounds tough, but once you have mastered the 10× table, it is just a few steps away. Firstly, 11× is mostly easy: from 11×2 to 11×9 you just put the two digits together. 11×2=22, 11×3=33, ..., 11×9=99. And of course 2×, 5× and 10× just follow their simple rules you know already. So it just leaves these to remember:Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. For math, science, nutrition, history ...Which expression is equivalent to log Subscript 12 Baseline StartFraction x Superscript 4 Baseline StartRoot x cubed minus 2 EndRoot Over (x + 1) Superscript 5 Baseline EndFraction? 4 log Subscript 12 Baseline x + one-half log Subscript 12 Baseline (x cubed minus 2) minus 5 log Subscript 12 Baseline (x times 1)Solve your math problems using our free math solver with step-by-step solutions. Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more.First we prove an intermediate result. Subtract 0 × 0 0 × 0 from each side to get 0 = 0 × 0 0 = 0 × 0. Now we are ready for the final kill. = 1 × 1 + 1 × (−1) + (−1) × 1 + (−1) × (−1) = 1 × 1 + 1 × ( − 1) + ( − 1) × 1 + ( − 1) × ( − 1) Add 1 1 to each side to get 1 = (−1) × (−1) 1 = ( − 1) × ( − 1). 25.6k 4 ... Which expression is equivalent to log Subscript 12 Baseline StartFraction x Superscript 4 Baseline StartRoot x cubed minus 2 EndRoot Over (x + 1) Superscript 5 Baseline EndFraction? 4 log Subscript 12 Baseline x + one-half log Subscript 12 Baseline (x cubed minus 2) minus 5 log Subscript 12 Baseline (x times 1)Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. For math, science, nutrition, history ... That you could view as x to the negative 1. You have a first power here. 1 minus 2 is negative 1. So this right here is equal to x to the negative 1 power. Or it could also be equal to 1 over x. These are equivalent. So let's say that this is equal into 1 over x, just like that. And it would be. x over x times x.Suppose that Am×nx = 0 has a nontrivial solution x. Prove that, for some row vector z, the equation yA = z has no solution. Assume for contradiction that yA =z has a solution y for all vectors z. Now, multiply on the right by x. This gives, yAx= zx⇒ y(0)= zx⇒ zx = 0. Then, since x is not the zero ...Symbolab is the best step by step calculator for a wide range of math problems, from basic arithmetic to advanced calculus and linear algebra. It shows you the solution, graph, detailed steps and explanations for each problem.For Question 1, observe that Z = (B×X)∩V. (Just notice that x ∈ V b means (b,x)∈ V .) Question 2: consider the map f:(B×C)×Pn → (B ×Pn)×(C ×Pn), (b,c,x) ↦((b,x),(c,x)). ... The problem is that in order to remedy the problems and paradoxes of naive set theory, the mathematicians around the turn of the century realised that you ...Associative property of multiplication: Changing the grouping of factors does not change the product. For example, (2 \times 3) \times 4 = 2 \times (3 \times 4) (2×3)×4 = 2×(3×4). Identity property of multiplication: The product of 1 1 and any number is that number. For example, 7 \times 1 = 7 7 ×1 = 7. Sep 2, 2012 · The numpy.repeat has been mentioned, and that's clearly the equivalent to what you want. But for completenes' sake, there's also repeat from the itertools standard library. . However, this is intended for iterables in general, so it doesn't allow repetions by index (because iterables in general do not have an index defin What is x times x equal to in algebra?To solve x multiplied by x, try to observe the pattern created by letting x be any number.After creating your list of n...Free Square Roots calculator - Find square roots of any number step-by-stepAlgebra. Multiply (x-5) (x-1) (x − 5) (x − 1) ( x - 5) ( x - 1) Expand (x−5)(x− 1) ( x - 5) ( x - 1) using the FOIL Method. Tap for more steps... x⋅x+x⋅ −1−5x−5⋅−1 x ⋅ x + x ⋅ - 1 - 5 x - 5 ⋅ - 1. Simplify and combine like terms. Tap for more steps... x2 − 6x+5 x 2 - 6 x + 5.Solve for x sin (x)=1. sin(x) = 1 sin ( x) = 1. Take the inverse sine of both sides of the equation to extract x x from inside the sine. x = arcsin(1) x = arcsin ( 1) Simplify the right side. Tap for more steps... x = π 2 x = π 2. The sine function is positive in the first and second quadrants. To find the second solution, subtract the ...In order to show that T is a function, you need to prove that for each pair of subsets A and B of X there is one and only one subset U of X\times X such that T(A,B)=U, that is, ((A,B),U) \in T ... What is the ''right" norm for the Banach space tensor product in this situation?Add, subtract, multiply and divide decimal numbers with this calculator. You can use: Positive or negative decimals. For negative numbers insert a leading negative or minus sign before your number, like this: -45 or -356.5. Integers, decimals or scientific notation. For scientific notation use "e" notation like this: -3.5e8 or 4.7E-9.Multiply (x-1) (x-1) (x − 1) (x − 1) ( x - 1) ( x - 1) Expand (x−1)(x− 1) ( x - 1) ( x - 1) using the FOIL Method. Tap for more steps... x⋅x+x⋅ −1−1x−1⋅−1 x ⋅ x + x ⋅ - 1 - 1 x - 1 ⋅ - 1 Simplify and combine like terms. Tap for more steps... x2 − 2x+1 x 2 - 2 x + 1Solve your math problems using our free math solver with step-by-step solutions. Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more. Max has a 5 \times 6 card and he enlarged it to be 7.5 \times 9. What was the scale factor? ... First lets forget about the decimal places and multiply 405 x 11. that ...Get the free "Solve for X Calculator" widget for your website, blog, Wordpress, Blogger, or iGoogle. Find more Mathematics widgets in Wolfram|Alpha.Solve your math problems using our free math solver with step-by-step solutions. Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more. Simplify x^ (1/2)*x^ (1/2) x1 2 ⋅ x1 2 x 1 2 ⋅ x 1 2. Multiply x1 2 x 1 2 by x1 2 x 1 2 by adding the exponents. Tap for more steps... x1 x 1. Simplify x1 x 1.Sep 2, 2012 · The numpy.repeat has been mentioned, and that's clearly the equivalent to what you want. But for completenes' sake, there's also repeat from the itertools standard library. . However, this is intended for iterables in general, so it doesn't allow repetions by index (because iterables in general do not have an index defin Associative property of multiplication: Changing the grouping of factors does not change the product. For example, (2 \times 3) \times 4 = 2 \times (3 \times 4) (2×3)×4 = 2×(3×4). Identity property of multiplication: The product of 1 1 and any number is that number. For example, 7 \times 1 = 7 7 ×1 = 7.Associative property of multiplication: Changing the grouping of factors does not change the product. For example, (2 \times 3) \times 4 = 2 \times (3 \times 4) (2×3)×4 = 2×(3×4). Identity property of multiplication: The product of 1 1 and any number is that number. For example, 7 \times 1 = 7 7 ×1 = 7. Identify H (A) and H (B) with H (pt). Then f ∗ = (g∗,−g∗), so kerf ∗ = kerg∗. Proof that Sorgenfrey plane is not normal using points x × (-x) They mean all x,−x +ϵ such that x ∈ (a,b) and 0 < ϵ < 1/n; that includes both rational and irrational x. You have a particular n and a non-empty open interval (a,b) ...You want to have your decimal q such that q \times (1.15 \times x ) = 1.15 \times x - x = 0.15 \times x so q \times 1.15 \times x = 0.15 \times x for all x Finally, you have q = \frac{0.15}{1.15} \simeq 0.1304347826 ...Free Algebraic Properties Calculator - Simplify radicals, exponents, logarithms, absolute values and complex numbers step-by-step Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor.You enter the first fraction, you enter the second fraction, click "Calculate" and hey presto, you get the answer. You can also click the little icon after the calculator to find out more information about the process of subtracting one fraction from another. (Note: you need to have performed a calculation first or the link won't work!)That you could view as x to the negative 1. You have a first power here. 1 minus 2 is negative 1. So this right here is equal to x to the negative 1 power. Or it could also be equal to 1 over x. These are equivalent. So let's say that this is equal into 1 over x, just like that. And it would be. x over x times x., the numerator is 3, and the denominator is 8. A more illustrative example could involve a pie with 8 slices. 1 of those 8 slices would constitute the numerator of a fraction, while the total of 8 slices that comprises the whole pie would be the denominator. If a person were to eat 3 slices, the remaining fraction of the pie would therefore beFree math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor.Apr 29, 2023 · An exponent is a way to represent how many times a number, known as the base, is multiplied by itself. It is represented as a small number in the upper right hand corner of the base. For example: x² means you multiply x by itself two times, which is x × x. Likewise, 4² = 4 × 4, etc. If the exponent is 3, in the example 5³, then the result ... Simplify 1/ ( square root of x) 1 √x 1 x. Multiply 1 √x 1 x by √x √x x x. 1 √x ⋅ √x √x 1 x ⋅ x x. Combine and simplify the denominator. Long Multiplication Example: Multiply 234 by 56. Long Multiplication Steps: Stack the numbers with the larger number on top. Align the numbers by place value columns. Multiply the ones digit in the bottom number by each digit in the top number. 6 × 4 = 24. Put the 4 in Ones place. Carry the 2 to Tens place., the numerator is 3, and the denominator is 8. A more illustrative example could involve a pie with 8 slices. 1 of those 8 slices would constitute the numerator of a fraction, while the total of 8 slices that comprises the whole pie would be the denominator. If a person were to eat 3 slices, the remaining fraction of the pie would therefore beA graduation cap is an example of a mapping cylinder g : X \to Y where X = S^1, Y = [-2,2] \times [-2,2], and g is the inclusion map. Verifying continuity of the deformation retraction of the mapping cylinderAll the constructions that you used to define the isomorphism are natural/functorial: Given a map X →Y, you have a natural map that respect inclusions, which gives a starting point for all the ... Let X =R. The homotopy will be from the identity map to itself, so H (0,x)= H (1,x)= x for all x. For each integer n ≥ 1, during the time period ... How to Use the Calculator. Type your algebra problem into the text box. For example, enter 3x+2=14 into the text box to get a step-by-step explanation of how to solve 3x+2=14.Solve your math problems using our free math solver with step-by-step solutions. Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more.Nov 12, 2018 · Add, subtract, multiply and divide decimal numbers with this calculator. You can use: Positive or negative decimals. For negative numbers insert a leading negative or minus sign before your number, like this: -45 or -356.5. Integers, decimals or scientific notation. For scientific notation use "e" notation like this: -3.5e8 or 4.7E-9. Simplify x^ (1/2)*x^ (1/2) x1 2 ⋅ x1 2 x 1 2 ⋅ x 1 2. Multiply x1 2 x 1 2 by x1 2 x 1 2 by adding the exponents. Tap for more steps... x1 x 1. Simplify x1 x 1. You enter the first fraction, you enter the second fraction, click "Calculate" and hey presto, you get the answer. You can also click the little icon after the calculator to find out more information about the process of subtracting one fraction from another. (Note: you need to have performed a calculation first or the link won't work!) The three integrals from 1 to 2, from 2 to 4, and from 4 to 8 are all equal. Each region is the previous region halved vertically and doubled horizontally. Extending this, the integral from 1 to 2 k is k times the integral from 1 to 2, just as ln 2 k = k ln 2. Calculus. In real calculus, the derivative of 1/x = x −1 is given by the power rule ... View solution steps Evaluate x Graph Quiz Polynomial x×1 Similar Problems from Web Search What does X × {0} and Y × {1} mean in this proof? https://math.stackexchange.com/q/2332997 A×B = {(a,b):a ∈ A,b ∈ B}. This is called the Cartesian product. For example {a,b,c}×{0,1} ={(a,0),(b,0),(c,0),(a,1),(b,1),(c,1)} In the context of the proof, ... Apr 28, 2022 · What is x times 1? Updated: 4/28/2022 Wiki User ∙ 11y ago Study now See answers (13) Best Answer Copy x times 1 is x. Anything times 1 is still that same anything. Wiki User ∙ 11y ago... To write 1 y 1 y as a fraction with a common denominator, multiply by x x x x. 1 x ⋅ y y + 1 y ⋅ x x 1 x ⋅ y y + 1 y ⋅ x x. Write each expression with a common denominator of xy x y, by multiplying each by an appropriate factor of 1 1. Tap for more steps... y xy + x xy y x y + x x y. Combine the numerators over the common denominator.Online math solver with free step by step solutions to algebra, calculus, and other math problems. Get help on the web or with our math app. For X\times [0,M]: \times is the Cartesian product and [0,M] is the interval \{x:0 \le x \le M\}. So, X \times [0,M] = \{(x,r):x \in X \land 0 \le r \le M\} For Y ...Free Square Roots calculator - Find square roots of any number step-by-stepA student was asked to prove the trigonometric identity tangent of one half times x plus cotangent of one half times x equals 2 times cosecant x period Which of the following could be the first step in proving the identity? the quantity 1 minus cosine x end quantity over sine x plus sin x over the quantity 1 minus cosine x end fquantity equals ...Defintion of proper homotopy https://math.stackexchange.com/questions/2532344/defintion-of-proper-homotopy Let X =R. The homotopy will be from the identity map to itself, so H (0,x)= H (1,x)= x for all x. For each integer n ≥ 1, during the time period [1/(n+1),1/n], the point n∈ X is ...Free Algebraic Properties Calculator - Simplify radicals, exponents, logarithms, absolute values and complex numbers step-by-step Step 1: Enter the radical expression below for which you want to calculate the square root. The square root calculator finds the square root of the given radical expression. If a given number is a perfect square, you will get a final answer in exact form. If a given number is not a perfect square, you will get a final answer in exact form and ...

By definition, (x,x)= {{x},{x,x}}. This last set is equal to {{x},{x}} ... Equivalence Relation, and finding the subset that defines the relation. Mostly right, which means wrong. The Transitive proof is correct. The symmetric proof is correct, but cluttered. You just have to say that: as multiplication of reals is commutative, then xy >0 .... V string

x times 1

Apr 12, 2019 · Long Multiplication Example: Multiply 234 by 56. Long Multiplication Steps: Stack the numbers with the larger number on top. Align the numbers by place value columns. Multiply the ones digit in the bottom number by each digit in the top number. 6 × 4 = 24. Put the 4 in Ones place. Carry the 2 to Tens place. Simplify x^ (1/2)*x^ (1/2) x1 2 ⋅ x1 2 x 1 2 ⋅ x 1 2. Multiply x1 2 x 1 2 by x1 2 x 1 2 by adding the exponents. Tap for more steps... x1 x 1. Simplify x1 x 1. 18 seconds. =. 55 seconds. Subtract minutes. 9 minutes is less than 56 minutes so borrow 1 from hours. There are 0 hours so borrow 1 from days. 1 day = 24 hours and 1 hour = 60 minutes, so add 24 to hours, then borrow 1 from hours to leave 23. Add 60 minutes to 9 to get 69. 69 minutes - 56 minutes = 13 minutes.Step 1: Enter the radical expression below for which you want to calculate the square root. The square root calculator finds the square root of the given radical expression. If a given number is a perfect square, you will get a final answer in exact form. If a given number is not a perfect square, you will get a final answer in exact form and ... The background is Munkres's topology says: Every closed interval in $\\mathbb{R}$ is compact. and A subspace A of $\\mathbb{R}^n$ is compact if and only if it is closed and is bounded in the square (orSolve your math problems using our free math solver with step-by-step solutions. Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more.Algebra. Multiply (x-5) (x-1) (x − 5) (x − 1) ( x - 5) ( x - 1) Expand (x−5)(x− 1) ( x - 5) ( x - 1) using the FOIL Method. Tap for more steps... x⋅x+x⋅ −1−5x−5⋅−1 x ⋅ x + x ⋅ - 1 - 5 x - 5 ⋅ - 1. Simplify and combine like terms. Tap for more steps... x2 − 6x+5 x 2 - 6 x + 5., the numerator is 3, and the denominator is 8. A more illustrative example could involve a pie with 8 slices. 1 of those 8 slices would constitute the numerator of a fraction, while the total of 8 slices that comprises the whole pie would be the denominator. If a person were to eat 3 slices, the remaining fraction of the pie would therefore be Get the free "Solve for X Calculator" widget for your website, blog, Wordpress, Blogger, or iGoogle. Find more Mathematics widgets in Wolfram|Alpha.First we prove an intermediate result. Subtract 0 × 0 0 × 0 from each side to get 0 = 0 × 0 0 = 0 × 0. Now we are ready for the final kill. = 1 × 1 + 1 × (−1) + (−1) × 1 + (−1) × (−1) = 1 × 1 + 1 × ( − 1) + ( − 1) × 1 + ( − 1) × ( − 1) Add 1 1 to each side to get 1 = (−1) × (−1) 1 = ( − 1) × ( − 1). 25.6k 4 ...Which expression is equivalent to log Subscript 12 Baseline StartFraction x Superscript 4 Baseline StartRoot x cubed minus 2 EndRoot Over (x + 1) Superscript 5 Baseline EndFraction? 4 log Subscript 12 Baseline x + one-half log Subscript 12 Baseline (x cubed minus 2) minus 5 log Subscript 12 Baseline (x times 1)In order to show that T is a function, you need to prove that for each pair of subsets A and B of X there is one and only one subset U of X\times X such that T(A,B)=U, that is, ((A,B),U) \in T ... 18 seconds. =. 55 seconds. Subtract minutes. 9 minutes is less than 56 minutes so borrow 1 from hours. There are 0 hours so borrow 1 from days. 1 day = 24 hours and 1 hour = 60 minutes, so add 24 to hours, then borrow 1 from hours to leave 23. Add 60 minutes to 9 to get 69. 69 minutes - 56 minutes = 13 minutes.Free Pre-Algebra, Algebra, Trigonometry, Calculus, Geometry, Statistics and Chemistry calculators step-by-stepDefintion of proper homotopy https://math.stackexchange.com/questions/2532344/defintion-of-proper-homotopy Let X =R. The homotopy will be from the identity map to itself, so H (0,x)= H (1,x)= x for all x. For each integer n ≥ 1, during the time period [1/(n+1),1/n], the point n∈ X is ...Like, what does “multiply ‘x’ by itself -1 times” mean? The expression x n only means “multiply x by itself n times” when n is a positive integer. When the exponent is 0, a negative integer, an arbitrary rational number, an arbitrary real number, or an arbitrary complex number you need a different definition for x n to make sense ....

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