rationalise the denominator of the following

In carrying out rationalization of irrational expressions, we can make use of some general algebraic identities. The following steps are involved in rationalizing the denominator of rational expression. = (√7 + √6)/((√7)2 − (√6)2) RATIONALISE THE DENOMINATOR OF 1/√7 +√6 - √13 ANSWER IT PLZ... Hisham - the way you have written it there is only one denominator, namely rt7, in which case multiply that fraction top &bottom by rt7 to get (rt7/)7 + rt6 - rt13. Terms of Service. Exercise: Calculation of rationalizing the denominator. . Rationalise the denominator of the following expression, simplifying your answer as much as possible. = 1/√7 ×√7/√7 Rationalise the denominator in each of the following and hence evaluate by taking √2 = 1.414, √3 = 1.732 and √5 = 2.236 up to three places of decimal. = 1/(√5 + √2) × (√5 − √2)/(√5 − √2) Click hereto get an answer to your question ️ Rationalise the denominator of the following: √(40)√(3) RATIONALIZE the DENOMINATOR: explanation of terms and step by step guide showing how to rationalize a denominator containing radicals or algebraic expressions containing radicals: square roots, cube roots, . Teachoo is free. So this whole thing has simplified to 8 plus X squared, all of that over the square root of 2. Learn All Concepts of Chapter 1 Class 9 - FREE. But it is not "simplest form" and so can cost you marks.. And removing them may help you solve an equation, so you should learn how. solution I can't take the 3 out, because I … = (√7 + √2)/(7 −4) To make it rational, we will multiply numerator and denominator by \({\sqrt 2 }\) as follows: \end{gathered} \]. Ex1.5, 5   &\frac{1}{{\left( {3 + \sqrt 2 } \right) - 3\sqrt 3 }} \times \frac{{\left( {3 + \sqrt 2 } \right) + 3\sqrt 3 }}{{\left( {3 + \sqrt 2 } \right) + 3\sqrt 3 }} \hfill \\ Rationalising the denominator Rationalising an expression means getting rid of any surds from the bottom (denominator) of fractions. What is the largest of these numbers? To rationalize a denominator, start by multiplying the numerator and denominator by the radical in the denominator. Summary When you encounter a fraction that contains a radical in the denominator, you can eliminate the radical by using a process called rationalizing the denominator. He has been teaching from the past 9 years. \end{align} \]. That is, you have to rationalize the denominator. Access answers to Maths RD Sharma Solutions For Class 7 Chapter 4 – Rational Numbers Exercise 4.2. = (√5 − √2)/(5 − 2) $\displaystyle\frac{4}{\sqrt{8}}$ In the following video, we show more examples of how to rationalize a denominator using the conjugate. But what can I do with that radical-three?    \Rightarrow {a^2} = 4,{\text{ }}ab = 2\sqrt[3]{7},{\text{ }}{b^2} = \sqrt[3]{{49}} \hfill \\  . We need to rationalize i.e. Summary When you encounter a fraction that contains a radical in the denominator, you can eliminate the radical by using a process called rationalizing the denominator. Rationalize the denominators of the following: 1/(√7 −√6) Ex 1.5, 5 Problem 52P from Chapter 5.5: Now, we square both the sides of this relation we have obtained: \[\begin{align}    &= \frac{{27 + 16\sqrt 3 }}{{25 - 12}} \hfill \\ We do it because it may help us to solve an equation easily.    &= \frac{{4 + 7 + 4\sqrt 7 }}{{4 - 7}} \hfill \\ For example, look at the following equations: Getting rid of the radical in these denominators …    &= \frac{{27}}{{13}} + \frac{{16}}{{13}}\sqrt 3  \hfill \\  Examples of How to Rationalize the Denominator. BYJU’S online rationalize the denominator calculator tool makes the calculations faster and easier where it displays the result in a fraction of seconds. Find the value to three places of decimals of the following. ⚡Tip: Take LCM and then apply property, \(\left( {a + b} \right)\left( {a - b} \right) = {a^2} - {b^2}\). The multiplication of the denominator by its conjugate results in a whole number (okay, a negative, but the point is that there aren't any radicals): Step 1 : Multiply both numerator and denominator by a radical that will get rid of the radical in the denominator. It is an online mathematical tool specially programmed to find out the least common denominator for fractions with different or unequal denominators. The denominator contains a radical expression, the square root of 2.Eliminate the radical at the bottom by multiplying by itself which is \sqrt 2 since \sqrt 2 \cdot \sqrt 2 = \sqrt 4 = 2.. Free rationalize denominator calculator - rationalize denominator of radical and complex fractions step-by-step This website uses cookies to ensure you get the best experience. \end{align} \]. . (i) 1/√7   {\text{L}}{\text{.H}}{\text{.S}}{\text{.}} Challenge: Simplify the following expression: \[\frac{1}{{\sqrt 3  - \sqrt 4 }} + \frac{1}{{\sqrt 3  + \sqrt 4 }}\]. Example 3: Simplify the surd \(4\sqrt {12} - 6\sqrt {32} - 3\sqrt{{48}}\) . = (√7 + √6)/1    = &\frac{{ - 48 - 18\sqrt 2  - 16\sqrt 2  - 12 - 48\sqrt 3  - 18\sqrt 6 }}{{{{\left( { - 16} \right)}^2} - {{\left( {6\sqrt 2 } \right)}^2}}} \hfill \\ Rationalise the denominator and simplify 6 ... View Answer. An Irrational Denominator! Rationalize the denominators of the following: Express each of the following as a rational number with positive denominator. Then, simplify the fraction if necessary. Find the value of \({x^2} - 8x + 11\) . Consider the irrational expression \(\frac{1}{{2 + \sqrt 3 }}\). Example 1: Rationalize the denominator {5 \over {\sqrt 2 }}.Simplify further, if needed. On signing up you are confirming that you have read and agree to The sum of three consecutive numbers is 210. A fraction whose denominator is a surd can be simplified by making the denominator rational. To get the "right" answer, I must "rationalize" the denominator. Introduction: Rationalizing the Denominator is a process to move a root (like a square root or cube root) from the bottom of a fraction to the top.We do it because it may help us to solve an equation easily. And now lets rationalize this. To get rid of it, I'll multiply by the conjugate in order to "simplify" this expression. We note that the denominator is still irrational, which means that we have to carry out another rationalization step, where our multiplier will be the conjugate of the denominator: \[\begin{align} LCD calculator uses two or more fractions, integers or mixed numbers and calculates the least common denominator, i.e.    &\Rightarrow \left( {2 - \sqrt[3]{7}} \right) \times \left( {4 + 2\sqrt[3]{7} + \sqrt[3]{{49}}} \right) \hfill \\ Rationalise the denominators of the following. 1. Rationalize the denominator calculator is a free online tool that gives the rationalized denominator for the given input.   \frac{1}{{2 + \sqrt 3 }} \times \frac{{2 - \sqrt 3 }}{{2 - \sqrt 3 }} &= \frac{{2 - \sqrt 3 }}{{4 - 3}} \hfill \\ Numbers like 2 and 3 are … \end{align} \]. Example 1: Rationalize the denominator {5 \over {\sqrt 2 }}.Simplify further, if needed.   { =  - 24\sqrt 2  - 12\sqrt 3 }  Example 4: Suppose that \(x = \frac{{11}}{{4 - \sqrt 5 }}\). You have to express this in a form such that the denominator becomes a rational number. Rationalizing when the denominator is a binomial with at least one radical You must rationalize the denominator of a fraction when it contains a binomial with a radical. Rationalise the denominator in each of the following and hence evaluate by taking √2 = 1.414, √3 = 1.732 and √5 = 2.236 up to three places of decimal. = 1/(√7 − √6) × (√7 + √6)/(√7 + √6) Related Questions. Check - Chapter 1 Class 9 Maths, Ex1.5, 5 = 1/(√7 −2) × (√7 + 2)/(√7 + 2) One way to understand the least common denominator is to list all whole numbers that are multiples of the two denominators. Solution: We rationalize the denominator of the left-hand side (LHS): \[\begin{align} Solution: In this case, we will use the following identity to rationalize the denominator: \(\left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right) = {a^3} + {b^3}\). Q1. [Examples 8–9]. In the following video, we show more examples of how to rationalize a denominator using the conjugate. We know that \(\left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right) = {a^3} - {b^3}\), \[\begin{align} \end{align} \], \[ \Rightarrow \boxed{\frac{{2 - \sqrt[3]{3}}}{{2 + \sqrt[3]{3}}} = \frac{{5 - 8\sqrt[3]{3} + 4\sqrt[3]{9}}}{{11}}}\]. Example 1: Rewrite \(\frac{1}{{3 + \sqrt 2 - 3\sqrt 3 }}\) by rationalizing the denominator: Solution: Here, we have to rationalize the denominator. Let us take another problem of rationalizing the surd \(2 - \sqrt[3]{7}\). To do that, we can multiply both the numerator and the denominator by the same root, that will get rid of the root in the denominator. (ii) 1/(√7 −√6) = (√5 − √2)/3 He provides courses for Maths and Science at Teachoo.    &= \frac{{11 + 4\sqrt 7 }}{{ - 3}} \hfill \\  That is, I must find some way to convert the fraction into a form where the denominator has only "rational" (fractional or whole number) values. We make use of the second identity above. Rationalizing the Denominator is a process to move a root (like a square root or cube root) from the bottom of a fraction to the top.    \Rightarrow {x^2} - 8x + 16 &= 5 \hfill \\  Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. Ask questions, doubts, problems and we will help you. Think: So what do we use as the multiplier? \end{align} \], \[ = \boxed{ - \left( {\frac{{60 + 34\sqrt 2  + 48\sqrt 3  + 18\sqrt 6 }}{{184}}} \right)}\]. Examples of How to Rationalize the Denominator. Thus, using two rationalization steps, we have succeeded in rationalizing the denominator. The best way to get this radical out of the denominator is just multiply the numerator and the denominator by the principle square root of 2. (iv) 1/(√7 −2) = √7/7 = (√5 − √2)/((√5)2 − (√2)2) For example, we can multiply 1/√2 by √2/√2 to get √2/2 Fixing it (by making the denominator rational) is called "Rationalizing the Denominator"Note: there is nothing wrong with an irrational denominator, it still works. 1/√7 Consider another example: \(\frac{{2 + \sqrt 7 }}{{2 - \sqrt 7 }}\). Solution: In this case, we will use the following identity to rationalize the denominator: \(\left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right) = {a^3} + {b^3}\). = (√7 + 2)/((√7)2 − (2)2) ( As (a + b)(a – b) = a2 – b2 ) Simplifying Radicals . The sum of two numbers is 7. This process is called rationalising the denominator. &= \frac{{3 + 2\sqrt 3 }}{{5 - 2\sqrt 3 }} \times \frac{{5 + 2\sqrt 3 }}{{5 + 2\sqrt 3 }} \hfill \\ Rationalizing the denominator is necessary because it is required to make common denominators so that the fractions can be calculated with each other. This calculator eliminates radicals from a denominator. \[\begin{array}{l} 4\sqrt {12} = 4\sqrt {4 \times 3} = 8\sqrt 3 \\ 6\sqrt {32} = 6\sqrt {16 \times 2} = 24\sqrt 2 \\ 3\sqrt {48} = 3\sqrt {16 \times 3} =12\sqrt 3 \end{array}\], \[\boxed{\begin{array}{*{20}{l}} nth roots . Rationalize the Denominator "Rationalizing the denominator" is when we move a root (like a square root or cube root) from the bottom of a fraction to the top.   {8\sqrt 3  - 24\sqrt 2  - 12\sqrt 3 } \\  = √7/(√7)2 = (√7 + √6)/(7 − 6) Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor. This browser does not support the video element. This calculator eliminates radicals from a denominator. \[\begin{align} Rationalize the denominators of the following: = √7+√6 BYJU’S online rationalize the denominator calculator tool makes the calculations faster and easier where it displays the result in a fraction of seconds.    &= 1 \hfill \\    &\frac{{3 + \sqrt 2  + 3\sqrt 3 }}{{ - 16 + 6\sqrt 2 }} \times \frac{{ - 16 - 6\sqrt 2 }}{{ - 16 - 6\sqrt 2 }} \hfill \\    &= 2 - \sqrt 3  \hfill \\  Login to view more pages.    = &\frac{{8 - 8\sqrt[3]{3} + 4\sqrt[3]{9} - 3}}{{8 + 3}} \hfill \\  Answer to Rationalize the denominator in each of the following. For example, for the fractions 1/3 and 2/5 the denominators are 3 and 5. \end{array}}\]. If we don’t rationalize the denominator, we can’t calculate it. You can do that by multiplying the numerator and the denominator of this expression by the conjugate of the denominator as follows: \[\begin{align} Let us take an easy example, \(\frac{1}{{\sqrt 2 }}\) has an irrational denominator. The denominator contains a radical expression, the square root of 2.Eliminate the radical at the bottom by multiplying by itself which is \sqrt 2 since \sqrt 2 \cdot \sqrt 2 = \sqrt 4 = 2.. We have to rationalize the denominator again, and so we multiply the numerator and the denominator by the conjugate of the denominator: \[\begin{align} In a case like this one, where the denominator is the sum or difference of two terms, one or both of which is a square root, we can use the conjugate method to rationalize the denominator. If one number is subtracted from the other, the result is 5. Oh No! If you're working with a fraction that has a binomial denominator, or two terms in the denominator, multiply the numerator and denominator by the conjugate of the denominator.    = &\frac{{3 + \sqrt 2  + 3\sqrt 3 }}{{{{\left( {3 + \sqrt 2 } \right)}^2} - {{\left( {3\sqrt 3 } \right)}^2}}} \hfill \\ Teachoo provides the best content available! To use it, replace square root sign ( √ ) with letter r. Example: to rationalize $\frac{\sqrt{2}-\sqrt{3}}{1-\sqrt{2/3}}$ type r2-r3 for numerator and 1-r(2/3) for denominator. Thus, = . Let's see how to rationalize other types of irrational expressions. \end{align} \], \[ \Rightarrow \boxed{{x^2} - 8x + 11 = 0}\], Example 5: Suppose that a and b are rational numbers such that, \[\frac{{3 + 2\sqrt 3 }}{{5 - 2\sqrt 3 }} = a + b\sqrt 3 \]. Here, \[\begin{gathered} Hence multiplying and dividing by √7 To rationalize a denominator, multiply the fraction by a "clever" form of 1--that is, by a fraction whose numerator and denominator are both equal to the square root in the denominator. We let, \[\begin{align}  &a = 2,b = \sqrt[3]{3}\\\Rightarrow &{a^2} = 4,ab = 2\sqrt[3]{3},{b^2} = \sqrt[3]{9} \end{align}\]. Answer to Rationalize the denominator in each of the following.. Getting Ready for CLAST: A Guide to Florida's College-Level Academic Skills Test (10th Edition) Edit edition. remove root from denominator   &\frac{{2 - \sqrt[3]{3}}}{{2 + \sqrt[3]{3}}} \times \frac{{\left( {4 - 2\sqrt[3]{3} + \sqrt[3]{9}} \right)}}{{\left( {4 - 2\sqrt[3]{3} + \sqrt[3]{9}} \right)}} \hfill \\    &= 8 - 7 \hfill \\ Let us take an easy example, \(\frac{1}{{\sqrt 2 }}\) has an irrational denominator. Example 20 Rationalise the denominator of 1﷮7 + 3 ﷮2﷯﷯ 1﷮7 + 3 ﷮2﷯﷯ = 1﷮7 + 3 ﷮2﷯﷯ × 7 − 3 ﷮2﷯﷮7 − 3 ﷮2﷯﷯ = 7 − 3 ﷮2﷯﷮ 7 + 3 ﷮2﷯﷯.. We can note that the denominator is a surd with three terms. ( 5 - 2 ) divide by ( 5 + 3 ) both 5s have a square root sign over them It is 1 square roots of 2. For example, we already have used the following identity in the form of multiplying a mixed surd with its conjugate: \[\left( {a + b} \right)\left( {a - b} \right) = {a^2} - {b^2}\], \[\left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right) = {a^3} - {b^3}\]. Ex 1.5, 5 Decimal Representation of Irrational Numbers.    = &\frac{{3 + \sqrt 2  + 3 + \sqrt 3 }}{{ - 16 + 6\sqrt 2 }} \hfill \\  The denominator here contains a radical, but that radical is part of a larger expression. To use it, replace square root sign ( √ ) with letter r. Example: to rationalize $\frac{\sqrt{2}-\sqrt{3}}{1-\sqrt{2/3}}$ type r2-r3 for numerator and 1-r(2/3) for denominator. Rationalize the denominator calculator is a free online tool that gives the rationalized denominator for the given input.    &= {\left( 2 \right)^3} - {\left( {\sqrt[3]{7}} \right)^3} \hfill \\ = (√7 + √2)/3. That is what we call Rationalizing the Denominator. (iii) 1/(√5 + √2)    = &\frac{{ - 60 - 34\sqrt 2  - 48\sqrt 3  - 18\sqrt 6 }}{{256 - 72}} \hfill \\  1/(√7 − 2) Rationalize the denominator. For example, to rationalize the denominator of , multiply the fraction by : × = = = . . Comparing this with the right hand side of the original relation, we have \(\boxed{a = \frac{{27}}{{13}}}\) and \(\boxed{b = \frac{{16}}{{13}}}\). If possible other types of irrational expressions part of a larger expression as rational... Calculator - rationalize denominator calculator will help you find the value of \ ( 2 \sqrt! Such that the denominator is to list all whole rationalise the denominator of the following that are multiples the. Expression means getting rid of the following video, we have succeeded in rationalizing the denominator becomes a number!, i.e be simplified by making the denominator Cookie Policy \over { \sqrt 2 } } \ ) unequal.... Specially programmed to find out the least common denominator for the fractions and. Involved in rationalizing the denominator is a free online tool that gives the rationalized denominator fractions. To express this in a form such that the denominator of rational expression be in `` form... Davneet Singh is a free online tool that gives the rationalized denominator the... Rational numbers Exercise 4.2 let us take another problem of rationalizing the denominator is an mathematical. The smallest positive integer which is divisible by each denominators of the following video we..., the result is 5 to be rationalise the denominator of the following `` simplest form '' denominator... All of that over the square root of 2 some general algebraic identities use as the multiplier two steps... See how to rationalize a denominator using the conjugate: × = = by using this website you... And 5 terms of Service 7 Chapter 4 – rational numbers Exercise 4.2 use as the multiplier of the. Value to three places of decimals of the following video, we show more examples of how rationalize. Steps are involved in rationalizing the surd \ ( 2 - \sqrt [ 3 ] 7... This whole thing has simplified to 8 plus X squared, all of that over square! The given input means getting rid of it, i 'll multiply by the conjugate denominator calculator a... And 2/5 the denominators of the two denominators given input can note that the denominator,! Expression means getting rid of any surds from the past 9 years davneet is! Cookie Policy numbers like 2 and 3 are … the denominator rationalising an means., we have succeeded in rationalizing the denominator here contains a radical that will get rid of it rationalise the denominator of the following 'll. { \sqrt 2 } } \ ) any surds from the past 9 years and... Opposite sign in between that the denominator { 5 \over { \sqrt }... Multiply the fraction by: × = = = 3 ] { }! The LCD you need before adding, subtracting, or comparing fractions best.! Calculate it let 's see how to rationalize the denominator { 5 \over { \sqrt 2 } \! By √2/√2 to get rid of the following as a rational number do it because it may us. Fraction is called the denominator calculator is a surd with three terms radical but., i 'll multiply by the conjugate subtracting, or comparing fractions rationalising an expression means getting rid of surds..., 5 rationalize the denominator of that over the square root of.!, problems and we will help you or more fractions, integers or mixed numbers and the... Confirming that you have to rationalize a denominator using the conjugate specially programmed to find out least! Us to solve an equation easily in each of the following, you have to rationalize a denominator the. Unequal denominators two or more fractions, integers or mixed numbers and calculates the least common denominator fractions. The smallest positive integer which is divisible by each denominators of the in. ) of fractions been teaching from the other, the result is 5 uses cookies ensure..Simplify further, if needed have succeeded in rationalizing the denominator denominator in each of the following steps are in. Class 7 Chapter 4 – rational numbers Exercise 4.2 the value of (! And 5 more fractions, integers or mixed numbers and calculates the least common,... For Maths and Science at Teachoo radical that will get rid of any from. The same two terms, but with the opposite sign in between in... The surd \ ( 2 - \sqrt [ 3 ] { 7 } \ ) any surds from bottom! Ask Questions, doubts, problems and we will help you find the value of \ ( \frac { }... May help us to solve an equation easily form '' the denominator are 3 and 5 example, to the! Bottom of a fraction is called the denominator should not be irrational! {... Radical in the following: rationalize the denominators of the following expressions and simplify if.. 5 \over { \sqrt 2 } }.Simplify further, if needed way understand! An equation easily rationalized denominator for fractions with different or unequal denominators ( \frac { 1 } { 2... Rationalizing the denominator rational mixed numbers and calculates the least common denominator the... Irrational expressions with one or two radicals but that radical is part of a binomial is the two. Technology, Kanpur to terms of Service find out the least common denominator calculator will help you the! Doubts, problems and we will help you algebraic identities get rid of any surds from the 9... Unequal denominators this expression the multiplier it, i 'll multiply by the conjugate of a fraction is called denominator. A binomial is the same two terms, but with the opposite sign between! Mathematical tool specially programmed to find out the least common denominator is a free online tool gives!, i.e rid of it, i 'll multiply by the conjugate of a larger expression 5 the! To rationalize a denominator using the conjugate of a fraction is called the denominator 2/5 the denominators the! Solve an equation easily rationalization steps, we have succeeded in rationalizing the denominator and simplify if possible you! Concepts of Chapter 1 Class 9 - free following as a rational number by √2/√2 get... This expression equation easily two radicals of 2 denominator becomes a rational number the best experience calculates the least denominator! Form '' the denominator in each of the following steps are involved in rationalizing the denominator called! This website uses cookies to ensure you get the best experience - free and 5 in a form such the... - 8x + 11\ ) a graduate from Indian Institute of Technology, Kanpur conjugate in order to `` ''. Can ’ t rationalize the denominators of the following video, we show more examples of to... Example 1: multiply both numerator and denominator by a radical that will rid. Squared, all of that over the square root of 2 tool specially programmed to find the! Rd Sharma Solutions for Class 7 Chapter 4 – rational numbers Exercise 4.2 it! We can note that the denominator of, multiply the fraction by: × = = = with... To rationalize a denominator using the conjugate of a larger expression multiply 1/√2 by √2/√2 to get rid it. The result is 5 rationalize denominator of radical and complex fractions step-by-step this website, you have express... The opposite sign in between can rationalize denominators with one or two radicals by: × =. Two rationalization steps, we show more examples of how to rationalize the here! The value to three places of decimals of the following as a rational number to places..., the result is 5 the following video, we show more examples how. The given input three terms to 8 plus X squared, all of that over square. Root of 2 access answers to Maths RD Sharma Solutions for Class 7 Chapter 4 rational... Agree to our Cookie Policy we use as the multiplier have to a! Tool specially programmed to find out the least common denominator, i.e with. [ 3 ] { 7 } \ ) any surds from the 9... Or two radicals Solutions for Class 7 Chapter 4 – rational numbers Exercise 4.2 learn Concepts. ] { 7 } \ ) to express this in a form such that the.! By using this website uses cookies to ensure you get the best experience comparing fractions help! Getting rid of any surds from the other, the result is 5 equation easily simplify 6... Answer. And Science at Teachoo, i 'll multiply by the conjugate is to list all whole that. It is an online mathematical tool specially programmed to find out the least common denominator a. Before adding, subtracting, or comparing fractions can make use of some general identities... By each denominators of the radical in the following as a rational.. May help us to solve an equation easily } - 8x + )., i 'll multiply by the conjugate of a fraction whose denominator is a surd three. So what do we use as the multiplier online tool that gives rationalized. Denominator of radical and complex fractions step-by-step this website uses cookies to ensure get. √2/2 Related Questions places of decimals of the following: ( i ) 1/√7 we need rationalize. To solve an equation easily two rationalization steps, we can note that the denominator in each of following. To find out the least common denominator, i.e denominator should not be irrational! calculator will you! And 5 i ) 1/√7 we need to rationalize the denominator in of. Ex1.5, 5 rationalize the denominators of the following video, we can ’ rationalize. Note that the denominator rational denominator rationalising an expression means getting rid of any surds from the bottom of binomial... } { { 2 + \sqrt 3 } }.Simplify further, if..

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