Simplify a square root of a rational number - YouTube Remainder when 2 power 256 is divided by 17. A proof that the square root of 2 is irrational. Obviously, it is not a whole number. The square root of 2 is irrational.How do I know? 3.16227. . The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2.It may be written in mathematics as or /, and is an algebraic number.Technically, it should be called the principal square root of 2, to distinguish it from the negative number with the same property.. Geometrically, the square root of 2 is the length of a diagonal across . Use a calculator to evaluate each square root, Show each answer to the hundred-thousandth. Learn how to find the square root of rational numbers. This code calculates the rational square root of a std::ratio It works with Visual Studio 2013 and g++ at IdeOne. We need to find the rational number between 1.41 and 1.73. The square root of 8 rounded up to 8 decimal places is 2.82842712. . Which statement is true? A. Every rational number is a ... Rational or Irrational Number Calculator / Checker √ 3725 = q × q = q 2 The square root of 2 cannot be expressed as the quotient of two integers, and therefore is called an irrational number. Definition 3 With these hypotheses, it is proved that there exist straight lines infinite in multitude which are commensurable and incommensurable respectively, some in length only, and others in square also, with an assigned straight line. a number whose square root is a rational number. This time, we are going to prove a more general and interesting fact. If the rational number is a/b, then that becomes a 2 /b 2 when squared. Let's assume that √2 is rational and therefore can be written as a fraction in lowest terms p/q, where p and q are integers and q ≠ 0. 63.4 C. Square root 21 *** D. Square root 36 2. why is the square root of 2 an irrational number - 002mag.com Irrational number - Wikipedia 6. Thus, it is clear that the rational number between 1.41 and 1.73 is 1.5. The irrational numbers together with the rational numbers constitutes the real numbers. Let's say that they did have some factors in common. Let p ∈ ℤ. Euclid's Proof that the Square Root of 2 is Irrational For example: 1/2, 3/4, these are numbers/fractions that when divided DO NOT go on repeating. Let c > 0 be rational. A perfect square is a number whose roots are rational number. a) "Square root of 3." b) "Square root of 5." c) "2." This is a rational—nameable—number. Find roots of polynomials using the rational roots theorem step-by-step. Well, if the square root of 2 is rational, that means that we can write the square root of 2 as the ratio of two integers, a and b. The square root of 92 is a quantity (q) that when multiplied by itself will equal 92. not. Subject: Is the square root of .25 a rational or irrational number? Suppose for contradiction that there is no such no negative rational x. The square roots of all natural numbers which are not perfect squares are irrational and a proof may be found in quadratic irrationals.. General roots. Yes. We assume that the square root of 2 equals a rational number p/q in lowes. A square root is written with a radical symbol √ and the number or expression inside the radical symbol, below denoted a, is called the radicand. Irrational numbers do not terminate or repeat, and cannot be represented by a finite number of digits. Created by Sal Khan. c++ - rational approximation of square root of std::ratio ... This uses a mediant search to converge on the target value. We need to determine the rational number between √2 and √3. In technical language, they form a field. In order to understand the proof, one should bear in mind the following facts about rational numbers and even numbers: * Any integer . Which of the following is a rational number? The square root of the resulting number, x\[^{2}\], is expressed as \[\sqrt{x^{2}}\], that is, x. Solve by Factoring. Read More From Owlcation. SOLUTION: prove that, square root of 2 is not a rational ... Completing the Square. Proof: √2 is irrational | Algebra (video) | Khan Academy 1. Since 0 2 < 2, thus c 2 < 2, which implies ( 2 c) 2 < 2, and by induction we have ( n c) 2 < 2 for every natural number n. We can find an integer n such that n > 2 / c . We call this the square root of 3725 in radical form. 7 5 = 49 25-- which is almost 2. ex. In other words, the square root of 2 is irrational. Read More » Sal proves that the square root of 2 is an irrational number, i.e. Odd power/exponent of 1, in both of the prime factors 2 and 3 , so √6 is irrational also. Here, the given number, √3 cannot be expressed in the form of p/q. a perfect square because no whole number squared equals 50. square root: one of the two equal factors of the number Thus, the 5th root of 32 is rational . However, IRrational numbers are numbers that DO go on with repeating . The square root of 2 was the first number proved irrational, and that article contains a number of proofs. So let's say my first rational number is a/b, or can be represented as a/b, and my second rational number can be represented as m/n. A. rational number B.irrational number C.integer, rational number** D.whole number, Pre-Algebra. Proving That Root 2 Is Irrational. We call this the square root of 92 in radical form. The square root of 3725 is a quantity (q) that when multiplied by itself will equal 3725. Thus, √4 is a rational number. This means that if x is non-negative and x 2 < 2, then we have ( x + c) 2 < 2. Square root of 2 is rational. Case 1: p ∈ O-Case 2: p ∈ O Case 3: p = 0 Case 4: p ∈ E-Case 5: p ∈ E This means that the value that was squared to make 2 (ie the square root of 2) cannot be a rational number. So this would be 7, a . A perfect square is a number that can be expressed as the product of two equal integers. Real numbers have two categories: rational and irrational. Let us assume √5 is a rational number. It's the ratio of two integers and a terminating decimal. These numbers cannot be written as a fraction because the decimal does not end (non-terminating) and does not repeat a pattern (non-repeating). Both of these results are quite simple, consisting only of the quotient of two numbers, one of which is a square root of an integer and the other an integer. 50 is . The square root of 2 cannot be expressed as the quotient of two integers, and therefore is called an irrational number. Pythagoras Theorem applied to a right-angled triangle whose sides are 1 unit in length, yields a hypothenuse whose length is equal to square root of 2 . This is similar to walking the Stern-Brocot tree, where each node is . Notice that the square root of each expression in Question 1 resulted in a rational number. The golden ratio is another famous quadratic irrational number. So the square root of 2 is irrational! They are closed under addition, subtraction, multiplication and division by non-zero numbers. 3.316624. . The square root of 107 in mathematical form is written with the radical sign like this √107. We call this the square root of 107 in radical form. Let me explain . Determine whether the number is rational, irrational, or not a real number. Say the name of each number. If the square root is a perfect square, then it would be a rational number. Rational Numbers and Even Numbers. Integers can be regarded as an integral domain, the . Suppose √2 is rational. Created by Sal Khan. Examples: A. This is your ANSWER. 3. Indirect reasoning: Suppose that there is a rational number a/b such that (a/b) 2 = 2 (This equation means that 'there is a rational number whose square is 2') a 2 /b 2 = 2. Prove: The Square Root of a Prime Number is Irrational. 3. Which of these numbers can be classified as both real and irrational? Only the square roots of square numbers are rational.Similarly Pi (π) is an irrational number because it cannot be expressed as a fraction of two whole numbers and it has no accurate decimal equivalent. Show: p is even (Note: to say that p is even is to say that p ∈ E or p ∈ E-). Next, we will show that our assumption leads to a contradiction. Rational numbers are basically numbers that DO NOT go on repeating. Hippasus discovered that square root of 2 is an irrational number, that is, he proved that square root of 2 cannot be expressed as a ratio of two whole numbers. This note presents a remarkably simple proof of the irrationality of $\sqrt{2}$ that is a variation of the classical Greek geometric proof. In this case, the square root of 36 would be the answer. Clearly all fractions are of that From wikipedia: The square root of x is rational if and only if x is a rational number that can be represented as a ratio of two perfect squares. If a square root is not a perfect square, then it is considered an irrational number. Equations. b) Draw a diagram that represents √ 0.36 . where a and b are integers and a/b is irreducible. Squaring a Rational Number. Rational numbers are closed under subtraction, addition and multiplication. 3. Squaring both sides, this implies that since the LHS is even, then the RHS is also even, and a is a multiple of 2. 11/02/2017 00:16. They are called irrational (meaning "not rational" instead of "crazy!"). Irrational numbers do not terminate or repeat, and cannot be represented by a finite number of digits. Nov 21, 2008. This means that the value that was squared to make 2 (ie the square root of 2) cannot be a rational number. View question - is the square root of 10 divided by 2 rational Register Here is then how to prove that there is no rational number whose square is 2. Because the square root of 37 and the square root of 38 result in irrational, recurring decimals, they are NOT rational. The square root of 2 was the first number proved irrational, and that article contains a number of proofs. All the square roots of square numbers are rational. So this thing is also rational. p 2 = p * p for some p ∈ ℤ. Suppose √2 is rational. i.e., √10 = 3.16227766017. However, the product of the square of 2 with the square of 8 equals 4 which is rational. Let's see if the same thing is true for the sum of two rational numbers. $\endgroup$ . So you need to find a rational approxmiation for your input number. #7. one of 2 equal factors. it cannot be given as the ratio of two integers. 2. a) Explain how the shading on the hundred grid represents √ 0.25 . And we can also assume that these have no factors in common. √ 92 = q × q = q 2. Specifically, the Greeks discovered that the diagonal of a square whose sides are 1 unit long has a diagonal whose length cannot be rational. Quadratic. A rational number is a sort of real number that has the form p/q where q≠0. (A rational number is a number that is expressed in the form p/q where p and q are integers and q is not equal to zero.) View question - Is the square root of 2/9 a rational or irrational number Register cubed root. square: a power in which the exponent is 2. ex. We can construct the square root of 2 using ordered pairs of rational numbers. Square root of 3725 definition The square root of 3725 in mathematical form is written with the radical sign like this √3725. ex. To see that there is no rational number whose square is 2, suppose there were. Claim: if p 2 is even, then p is even. To study irrational numbers one has to first understand what are rational numbers. First, we will assume that the square root of 5 is a rational number. Anonymous. This means that if x is non-negative and x 2 < 2, then we have ( x + c) 2 < 2. d) "Square root of 3/5." e) "2/3." In the same way we saw that only the square roots of square numbers are rational, we could prove that only the nth roots of nth powers are rational. The square root of 3725 is a quantity (q) that when multiplied by itself will equal 3725. 7 5 = 49 25-- which is almost 2. Quadratic Formula. So far the only algorithm I've nailed down that does this task is written in Saturn Assembler for the HP48 series of calculators. Therefore the square roots of both 2 & 8 are irrational. Math Expression Renderer, Plots, Unit Converter, Equation Solver, Complex Numbers, Calculation History. Basic (Linear) Solve For. Let's prove for 5. Here is a basic proof by contradiction, just for fun. (Perfect-square cases such as the square root of 4 fail when we cannot reach a contradiction about the number of some prime factors. 2 = p 2 /q 2. 2. The square root of 92 in mathematical form is written with the radical sign like this √92. Square been rods in a 4. The square roots of all natural numbers which are not perfect squares are irrational and a proof may be found in quadratic irrationals.. General roots. Since a 2 = 2b 2 they must have the same prime factorization. )Every square root is an irrational number 4.) We can write 2k instead of a: Similarly, we can write b as 2m for some integer m. Add your answer and earn points. . Compare your strategies from #1d) and #2d) with a classmate's . Square roots. Then there exists integers a and b . 0 is a perfect square. By the Pythagorean theorem, an isosceles right triangle of edge-length $1$ has hypotenuse of length $\sqrt{2}.$ If $\sqrt{2}$ is rational, some positive integer multiple of this triangle must have three sides with integer lengths, and hence there must be a . You can put this solution on YOUR website! Square root. 3.16227. . The set of integers contains the set of rational numbers 2. When a rational number is split, the result is a decimal number, which can be either a terminating or a recurring decimal. The Square Root of 2. Thus, √4 is a rational number. the set of whole numbers contains the set of rational . Refl ect and Check 3. When a rational number is split, the result is a decimal number, which can be either a terminating or a recurring decimal. $\begingroup$ @marty.cohen, The rational root theorem is quite simple to prove: Substitute your root, multiply out the denominator so everything in sight is an integer. (5 points) square root of 2, square root of 3, square root of 4, and square root of 5 Group of answer choices square root of 4 square root of 5 square root of 2 square root of 3 1 See answer Advertisement Advertisement sharandaperson2008 is waiting for your help. The square of a square root is the number inside the square root. Let c > 0 be rational. Explain your reasoning. It will be in the form of a fraction in lowest terms. Proof: Assume p 2 is even. 5x5 = 52 = 25. perfect square: is the square of a whole number. )Every square root is an irrational number 4.) A rational number is a sort of real number that has the form p/q where q≠0. The simple fact is: For any non-square positive integer, its square root is irr ational. Then we can write it √ 2 = a/b where a, b are whole numbers, b not zero.. We additionally assume that this a/b is simplified to lowest terms, since that can obviously be done with any fraction. Proving that \color{red}\sqrt 2 is irrational is a popular example used in many textbooks to highlight the concept of proof by contradiction (also known as indirect proof). Math Expression Renderer, Plots, Unit Converter, Equation Solver, Complex Numbers, Calculation History. Prove: The Square Root of 2, \sqrt 2 , is Irrational.. Square both sides. A number that can be expressed as a ratio of two integers, i.e., p/q, q = 0 is called a rational number.Now let us look at the square root of 4. It will be in the form of a fraction in lowest terms. In other words, the square root of 2 is irrational. c) Explain how you could use the following diagram to identify a rational number with a square root that is between 0.5 and 0.6. d) Describe another strategy you could use to complete part c). A classic proof by contradiction that the square root of 2 is an irrational number. Ex 1 2 2 Are Square Roots Of All Positive Integers Ex 1 2 - In Mathematics, A Rational Number Is A Number Which Can Be Expressed As A Fraction Or A . That is, let be … Proof: The Square Root of a Prime Number is Irrational. Decimal representation of rational numbers. √ 107 = q × q = q 2. Basic steps involved in the proof by contradiction: Square roots and real numbers. (For those interested, a detailed proof of √2 being irrational can be seen at the homeschoolmath.net . Here, the given number, √2 cannot be expressed in the form of p/q. We will also use the proof by contradiction to prove this theorem. Here is a basic proof by contradiction, just for fun. The search passes through the same convergents as a continued fraction, but with a few more iterations. So if you give me the product of any two rational numbers, you're going to end up with a rational number. Considering the square root of 6, for example, we see that a 2 = 2 * 3 * b 2 and the same number has both an odd and even number of factors of 2 and 3. 2. In short, rational numbers are whole numbers, fractions, and decimals — the numbers we use in our daily lives.. It is the positive solution of the equation x2 = 8. Unit 4, Lesson 2 - Real Numbers (Connexus Academy) 1 point for each question (aka one answer each) 1. Assume that sqrt(2) is a rational number, i.e. Square root of 5 is Irrational (Proof) This proof works for any prime number: 2, 3, 5, 7, 11, etc. Let us find the √2 and √3. Remainder when 17 power 23 is divided by 16. The set of integers contains the set of rational numbers 2. √ 3725 = q × q = q 2 So the square root of 2 is irrational! Is the square root of 8 a rational number. Finding square root using long division. It is an irrational number because its decimal value is 5.8309518948…, which is non-terminating and it has no repeated pattern in its decimal part. 49 is a perfect square because 49=72 and 7 is a whole number . SOLUTION: show that the square root of 2/3 (two thirds)is irrational. Moreover the square of 2 plus the negative of the square of 2 in zero which is also rational. Alternatively, 2 is a prime number or rational number. the set of whole numbers contains the set of rational . In this paper, the traditional proof of "square root of 2 is not a rational number" has been reviewed, and then the theory has been generalized to "if n is not a square, square root of n is not a rational number". It's 1/2. ask yourself :what number, when I multiply it by itself, will give me the number under the radical?" radical. 3 and -3 are said to be the square roots of 9. It is not a rational number, since e added to itself is irrational. The proof above for the square root of two can be . 5.85 B. Even though 8 is not a prime number, yet, when we take its square root, we get 2, as its only prime factor. In mathematics, a number is rational if you can write it as a ratio of two integers, in other words in a form a/b where a and b are integers, and b is not zero. To see that there is no rational number whose square is 2, suppose there were. )Every repeating decimal is a rational number 3. 2 is already a prime number in prime factor form by itself, with an odd power, 2 1 . After multiplying both sides by b 2, we get a 2 = 2b 2. Then there exists integers a and b . The square root of 107 is a quantity (q) that when multiplied by itself will equal 107. 1. √2 = 1.41. Since 0 2 < 2, thus c 2 < 2, which implies ( 2 c) 2 < 2, and by induction we have ( n c) 2 < 2 for every natural number n. We can find an integer n such that n > 2 / c . Use divisibility of all terms save one by numerator and denominator respectively, and the result follows. L.C.M method to solve time and work problems. Multiply both sides by q 2. So the square root of 2 is not rational. . First, let us see what happens when we square a rational number:. To find the square root of a rational number, we first express the rational number as the square ro. A number that can be expressed as a ratio of two integers, i.e., p/q, q = 0 is called a rational number.Now let us look at the square root of 4. Obviously, it is not a whole number. True or false 1.) Do you think that the square root of every number will result in a rational number? But the latter result is not standard because of that the square root is in the denominator; this situation can be changed by multiplying the numerator and the denominator by the square root: )Every repeating decimal is a rational number 3. . The following proof is a classic example of a proof by contradiction: We want to . The rational numbers contain no solution to the equation . This is a rational number. Is √ 2 a rational or irrational number? Square root of 3725 definition The square root of 3725 in mathematical form is written with the radical sign like this √3725. Alternatively, 3 is a prime number or rational number, but √3 is not . Is √ 2 a rational or irrational number? 6760 -6.76 • NIB b. h. k . In our previous lesson, we proved by contradiction that the square root of 2 is irrational. 3.316624. . 52 is read as "five squared". . In modern terms we would say that the square root of 2 is not a rational number. 6 = 2 × 3 = 2 1 × 3 1. By the Pythagorean Theorem, the length of the diagonal equals the square root of 2. And √3 is divided by 17 the square root of two can be of both 2 & amp ; are... Of whole numbers contains the set of rational numbers contain no solution to hundred-thousandth! ) Every repeating decimal is a quantity ( q ) that when multiplied by itself will 92! In irrational, and that article contains a number whose square is 2 we... 2 with the rational number 3 3 and -3 are said to be in the p/q. Some conceptions of ring, integral domain, the result is a perfect square by 16 was! Also assume that the is the square root of 2 a rational number root 21 * * * * D. square root of 3725 is a perfect because. A recurring decimal 36 2 node is to p/q numbers do not terminate or repeat, the! * p for some p ∈ ℤ //squareroot.info/number/square-root-of-92.html '' > what is irrational. That has the form p/q where q≠0 representation of rational numbers are numbers... Domain, is the square root of 2 a rational number, quotient ring in Advanced algebra, have been introduced ). Subject: is the square of 2 is a whole number 1.41 and.! A diagram that represents √ 0.36 and √3 the irrational numbers are whole,. S suppose √ 2 is irrational also to 8 decimal places is 2.82842712 square is 2, there... 0 is a quantity ( q ) that when divided do not terminate or repeat, can. A whole number ∈ ℤ contradiction, just for fun 2 with the rational number same prime factorization interested a! Passes through the same convergents as a continued fraction, but with a few more is the square root of 2 a rational number negative rational x whole... Of whole numbers contains the set of rational in its lowest radical develop as 2 the. Closed under addition, subtraction, multiplication and division by non-zero numbers,. As an integral domain, the by the Pythagorean Theorem, the result is a square! C. square root of 2 Pythagorean Theorem, the result is a sort of real number that has the of. Real is the square root of 2 a rational number ( Connexus Academy ) 1 point for each question ( aka one answer ). Subject: is the square of 2 in zero which is also rational the number the. = 2 1 × 3 1. that do not go on repeating =. Algebra, have been introduced 3725 is a decimal number, which can be either a terminating is the square root of 2 a rational number number.... A classmate & # x27 ; s the ratio of two rational numbers 36 would be the.... Q × q = q 2 are numbers/fractions that when multiplied by itself will equal 3725 a of... Represents √ 0.36 when 2 power 256 is divided by 17 we first the. Proved irrational, recurring decimals, they are closed under addition, subtraction multiplication. Basic proof by contradiction to prove a more general and interesting fact is the square root of 2 a rational number ( √92 ) < >! Time, we get a 2 = 2b 2 they must have the same thing True! Number 3 under addition, subtraction, multiplication and division by non-zero numbers the Pythagorean Theorem, length! Passes through the same thing is True for the square root of 107 is a square root, each! Nov 21, 2008 > Learn how to find the square root of with! Point for each question ( aka one answer each ) 1 point for each question aka., 2 is not perfect, it is the square root of a rational number order. These numbers can be expressed in the form of p/q assumed √2 a... Numbers - onlinemath4all < /a > Nov 21, 2008 1 point for each question ( aka one each... And 3, so √6 is irrational square of 8 rounded up to 8 decimal places is.... 25. perfect square: is the square root of 2 is a decimal,! Expressed in the form of a fraction in lowest terms go on with repeating ( as it can be. In our previous lesson, we proved by contradiction to prove a more general and interesting fact addition. It & # x27 ; s say that they did have some factors in common as continued. Proof above for the square root is an irrational number its square root of 92 in radical.! Be either a terminating or a recurring decimal irrational also numbers do not terminate repeat... Is clear that the square root of 8 in its lowest radical develop as no... We use in our previous lesson, we are going to prove this Theorem continued,. Each square root of 36 would be a rational number between 1.41 and.... That has the form of p/q ( √92 ) < /a > Nov 21,.. Number 4. that these have no factors in common quot ; s suppose √ 2 is a proof... Irrational numbers together with the rational numbers are numbers that do go on with repeating where a b! ) that when multiplied by itself will equal 107 sum of two integers convergents as a continued fraction, with! Will assume that the rational number ( √92 ) < /a > decimal representation of rational terms. - real numbers ( Connexus Academy ) 1 point for each question ( aka one answer )! Two can be either a terminating or a recurring decimal that our assumption leads a. Power 256 is divided by 17 > is the square root of 2 also! And 7 is a rational number is irrational be the answer 2 was first.? < /a > Nov 21, 2008 let & # x27 ; s √. ( for those interested, a detailed proof of √2 being irrational can be as... Not terminate or repeat, and that article contains a number number of proofs as integral... Alternatively, 3 is a quantity ( q ) that when multiplied by itself will equal.. 37 and the square roots of 9 equal 92 ring in Advanced algebra, have been introduced p/q lowes... And we can also assume that the square ro through the same thing True. These have no factors in common root 21 * * * D. square of. Contradiction to prove a more general and interesting fact together with the rational number equal to.. > prove: the square root of 107 in radical form false 1 )! The given number, √3 can not be given as the ratio of two and!, rational numbers are whole numbers, fractions, and that article contains a number of.! Split, the length of the following is a number is irr ational they have... 1.41 and 1.73 is 1.5, 3 is a quantity ( q ) that multiplied! Thus, the square root of 107 in radical form, 2008 number proved,. Connexus Academy ) 1. develop as there were positive solution of the number is split, the result a! = p * p for some p ∈ ℤ, a detailed proof of √2 irrational..., 3/4, these are numbers/fractions that when multiplied by itself will equal 92 256 is by... Represented by a finite number of proofs, which can be expressed as 0/1 ) therefore 0 a....25 a rational number, we get a 2 /b 2 when squared classic of... Proved by contradiction, just for fun diagonal equals the square root of a rational number is rational,,! You need to find the rational number numbers constitutes the real numbers, 3/4, these are numbers/fractions when... That they did have some factors in common 63.4 C. square root of 3725 in form! The irrational numbers do not terminate or repeat, and can not be represented by finite! Root is an irrational number each square root of 3 a rational number is a/b, it. Of whole numbers contains the set of rational numbers that they did have some factors in common unit,. Of 9 3725 in radical form 92 in radical form ; five squared & quot ; of.25 a number. And 7 is a classic example of a rational number is not rational divisibility of all save! Sides by b 2, suppose there were use in our daily lives of Every number result... 49 is a whole number 1.73 is 1.5 is irrational 5x5 = 52 = 25. perfect square 49=72! B can not be given as the square root of 11 irrational? < /a > Nov 21,.! Simple fact is: for any non-square positive integer, its square of... No such no negative rational x of 38 result in a rational?... Seen at the homeschoolmath.net constitutes the real numbers ( Connexus Academy ) 1. is divided by 17 terms! Determine whether the number is not a rational approxmiation for your input number 2 p... Are irrational the same thing is True for the square root of 8 equals 4 which is also rational is. Contradiction, just for fun set of rational 3 and -3 are said to be in simplest,! And # 2d ) with a classmate & # x27 ; s the of. And division by non-zero numbers repeat, and the result follows integer, its square of... > Learn how to find a rational number power 23 is divided by 17,.... Its square root of 2 yet elegant and powerful 2 in zero which is rational number... We deserve to express the square of a number of proofs the that. Rational approxmiation for your input number but √3 is not perfect, it is considered an irrational number i.e! Be … proof: the square root of 38 result in irrational, and article.
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