Step 1: This is a right triangle with two equal sides so it must be a 45-45-90 triangle. In other words, 3:4:5 refers to a right triangle with side length of 3, 4, and 5, where the hypotenuse is the length of 5 and the legs are 3 and 4, respectively. (HINT: Order Matters!). How to simplify square roots Square roots are radicals of order 2. Q: How to use pythagorean theorem with only one side? The principle square root is simply the radical sign. In other words, we need to compute 26 + 464, 26 464, and 26 / 464. We begin with the sum: 26 + 464. Step-by-step explanation: The given triangle is a right-angled triangle. And with a 30-60-90, the measure of the hypotenuse is two times that of the leg opposite the 30 angle, and the measure of the other leg is sqrt(3) times that of the leg opposite the 30 also seen in the diagram below. Rationalize radical denominator. (This works since a number divided by itself is 1. Since the larger square has sides c and area c2, the above can be rewritten as: which is again, the Pythagorean equation. A: To find the measure of side Q: _3. Search our database of more than 200 calculators, calculator works with decimals, fractions and square roots (to input $ \color{blue}{\sqrt{2}} $ type $\color{blue}{\text{r2}} $). Side 3=x Step 2: Click the blue arrow to submit. a x b = (a x b) radical-equation-calculator. The given triangle is right triangle. Let us take this opportunity to mention a couple of essential rules that govern Omni's simplify radicals calculator. The digit sum calculator helps you calculate the total sum of the digits in the given numbers. We pull the numbers representing the groups of four out of the radical and keep the rest inside. Answer (1 of 8): Note: this answer is kind of long so jump to picture to see a visual process. https://en.wikipedia.org/wiki/Pythagorean_theorem, https://www.calculatorsoup.com/calculators/geometry-plane/pythagorean-theorem.php. We find the prime factorization of the number under the root: and look for fours of the same primes. We will shortly describe how to simplify the radical expressions given by each of them. Or what do we do if it's 390,624 instead? Side-based right triangles figures that have side lengths governed by a specific rule, e.g. Express the answer in simplest radical form. Assuming Capital letters are angles, opposite legs are lower case In triangle XYZ, cosX = 1 / 2, cosY = 1 / 4 and x = 6. Let us focus on such expressions for the remainder of this section, so for now, you can consider our tool as a simplify square roots calculator. This calculator solves the Pythagorean Theorem equation for sides a or x = 4, y = 4 Road Signs The moose warning sign at the left is an equilateral triangle.The height of the sign is 1 m. Find the length s of each side of the sign to the nearest tenth of a meter. c. The hypotenuse is the side of the triangle opposite the right angle. Wolfram Demonstrations Project & Contributors | Terms of Use | Privacy Policy | RSS Let's start with the sum. It's the only possible right triangle that is also an isosceles triangle. That doesn't stop your teacher from giving the whole class a task to try to do on your own while they sit behind their desk with a phone in hand. Direct link to Gabi Cheng's post what happens if the two r, Posted 7 years ago. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. Type in the given value. We see that there is none, so that summand is already in its simplest form. xref 5) 15 yd (6) 8 km 13 yd h-9- C 16 km Find the missing side of each right triangle. Welcome to MathPortal. The given parameters are: Triangle type = Equilateral triangle. fractions simplified 5th grade. Select the triangle you need and type the given values the remaining parameters will be calculated automatically. vidDefer[i].setAttribute('src',vidDefer[i].getAttribute('data-src')); 0000003348 00000 n Convert to Radical Form, Convert to Radical Form (5x)-12 And once we apply it, we get something that returns us to point 1., and simplifying such radical expressions is no biggie. Our simplest radical form calculator uses all these properties to reduce your expression to the easiest possible. If you're not yet familiar with this idea, discover it with our prime factorization calculator. Step-by-step explanation: The following data were obtained from the . A Pythagorean triple is a set of 3 positive integers for sides a and In other words, while the exponent turns 5 into 58, the (eighth) root makes 58 into 5. Find the missing side of the triangle. Cite this content, page or calculator as: Furey, Edward "Pythagorean Theorem Calculator" at https://www.calculatorsoup.com/calculators/geometry-plane/pythagorean-theorem.php from CalculatorSoup, 0000006756 00000 n defenition of measurement for 1st grade. This website's owner is mathematician Milo Petrovi. Direct link to Richard Johnson's post How did you know right of, Posted 10 years ago. 13 mi 4 mi x Or while cleaning the house? We're interested in adding two values, so we choose "sum" under "Expression." Please provide any 2 values below to solve the Pythagorean equation: a2 + b2 = c2. Using the sides, find the distance between the two points in simplest radical form. Let's have a look at the example: we want to find the length of the hypotenuse of a right triangle if the length of one leg is 555 inches and one angle is 4545\degree45. For the product and quotient, we repeat the above steps (the a, b, and so on don't change), but choose the correct option under "Expression" product and quotient, respectively. For more on this special ratio, head to our golden ratio calculator. 0000006998 00000 n The free calculator will solve any square root, even negative ones and you can mess around with decimals too!The square root calculator below will reduce any square root to its simplest radical form as well as provide a brute force rounded approximation of any real or imaginary square root.. To use the calculator simply type any positive or negative number into the text box. How to Simplify Radicals (NancyPi) How To Write Mixed Radicals as Entire Radicals The How-To Scholar: Education Evaluating nth Roots and Rational Exponents Marty Brandl 22K views Adding and. Leave your answer in simplest radical form. var vidDefer = document.getElementsByTagName('iframe'); Additionally, you will discover why it's very important on how you choose your side lengths. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Of course, the most important special right triangle rule is that they need to have one right angle plus that extra feature. %PDF-1.4 % The sum of the area of these four triangles and the smaller square must equal the area of the larger square such that: In the second orientation shown in the figure, ii, the four copies of the same triangle are arranged such that they form an enclosed square with sides of length b - a, and area (b - a)2. // Last Updated: January 21, 2020 - Watch Video //. The special right triangle 3030\degree30 6060\degree60 9090\degree90 is one of the most popular right triangles. -x^2 - 7x = -11; Find the value of x. This time, the rules of mathematics give (a b) / (c d) = (a / (c d)) (b d) with k = lcm(n,m), s = k / n, and t = (k (m - 1)) / m. The extra m - 1 factor in t comes from rationalizing the denominator. c = hypotenuse Q: How to find the hypotenuse in special right triangles? Their areas are in geometric progression, according to the golden ratio. b. From Pythagorean Theorem we know that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs. 26 + 464 = 26 + 2 4(22) = 26 + 22. Read More simplify. Is every square root aksing for the principal square root? Nevertheless, there are some nifty tricks that we can use, and you bet we will show you all of them! Generally, special right triangles may be divided into two groups: Angle-based right triangles for example 3030\degree30-6060\degree60-9090\degree90 and 4545\degree45-4545\degree45-9090\degree90 triangles. The most important rule here is that we can only add radicals with the same order and number under the root symbol, i.e., we must have n = m and b = d. Otherwise, we can't add the two. As opposed to point 2., here, there's no need to look at each factor separately. You can specify conditions of storing and accessing cookies in your browser. This results in the formation of a larger square with sides of length b + a, and area of (b + a) 2. Wow! If all the primes under the radical have powers with a common factor with the root's order, we can reduce them in a similar way we reduce fractions to obtain an equivalent one. 0000002926 00000 n Well, sometimes, instead of approximating the result, it's better to transform it a little. Use the pythagorean theorem to find the missing side and writing the answer in simplest radical form It follows that the length of a and b can also be determined if the lengths of the other two sides are known using the following relationships: The law of cosines is a generalization of the Pythagorean theorem that can be used to determine the length of any side of a triangle if the lengths and angles of the other two sides of the triangle are known. example 3: Get access to all the courses and over 450 HD videos with your subscription. Enter the number under the radical you want to simplify. Find the value of each variable. Special Right Triangles with Radicals In the video below, you will also explore the 30-60-90 triangle ratios and use them to solve triangles. kn (aks bkt cku dkv ) = n(as bt cu dv ). In order to do this, you take a number and divide it by different prime numbers until all of its factor. Similar to point 3. Additionally, there are times when we are only given one side length, and we are asked to find the other two sides. Solve the equation. AboutTranscript. Unfortunately, this is not over yet. Provide any two values of a right triangle calculator works with decimals, fractions and square roots (to input type ) leg = leg = hyp. We pull these out of the radical and get: 26 / 464 = 0.03125 49,437,184 = 0.03125 4(220 32) = 0.03125 25 4(32) = 4(32). Loe can paint 27 feet of fence in 4.5 hours. Well, how fortunate that we have Omni's simplifying radicals calculator at hand! Jenn, Founder Calcworkshop, 15+ Years Experience (Licensed & Certified Teacher). Example 2: It's formed by three square sides. In the diagram below of right triangle ACB, altitude CD intersects AB at D. If AD =3 and DB = 4, A: Consider the triangle ABC let the value of the CAB is x, then the value of the CBA is calculated Q: a right triangle with right angle C. Side AC is 2 units longer than side BC. Right triangle, the sides of which are in a geometric progression (Kepler triangle). This calculator also finds the area A of the right triangle with sides a and Next, we recall that the root at hand is of order 2, so we check how many pairs of the same primes we obtained in the factorization: we have two pairs of 2s, a pair of 3s, and a single 2 is left alone. Pythagorean Theorem. The four triangles with area. This is a 30-60-90 triangle with one side length given. Express your answer in simplified radical form. Why don't we prove them wrong? In the second summand, we see that the order of the root and the powers of all (i.e., of the only one) numbers under it have a common factor 2. Indeed, there is one such solution (four 2s), which leaves two 2s alone. Referencing the above diagram, if. The length of side a is the square root of the squared hypotenuse minus the square of side h^2 = 10^2 + 6^2 = 100 + 3 = 136 h = 136 they expect you to reduce 136 to a radical expression in lowest terms 136 = 4 * 34 = 234 k can only be reduced if k contains one or more factors which are perfect squares, e.g b = side leg b That gives us a final answer of: Arguably, it took some time to write all the details, but the operations themselves weren't too terrible. Can we f, We're asked to add and simplify and we have the principle root of two x squared plus four times the principle root of eight plus three times the principle root two x squared plus the principle root of eight so we can do a little bit of adding, we can actually simplify first and then add or we can add first and then simplify but it looks like we can already add so lets try and do that so here, right over here, I have a principle root of two x squared and over here I have three principle roots of two x squared, well if I have one of something here and I have three of something here and i need to add them together I can put a one co-efficient out here to make it clear this is one of this thing and I have three of these things but if I have one of this thing and three more of these things and I add them together I am going to have four of those things, so this is four times the principle root of two x squared and that confuses a little bit, imagine that the whole principle root of two x squared was some variable lets say this whole thing was "a" and lets say that this whole thing was "a" as well, because its the same thing, you'd have one "a" plus three "a"'s which will give you four "a"'s, in this case "a" is all of this business right over here so we added those terms, and then we wanted to think about we have four principle roots of "a" and we have one more principle roots of "a", so same idea you have four of these things I am circling in magenta and you have one more of these things that I am circling in magenta, that one co-efficient is implicit so if I have four of something plus one more of something it becomes five of that something so plus plus five times the square root, plus five times the square root of eight and now we'll see if we can simplify this anymore, we have four of something and we have five of something else, so you can't just add these two things together, but maybe we can simplify this a little bit so we know that the principle root of two x squared, this is the same thing as, so let me write the four out front, so we have the four, and the principle root of two x squared is the same thing as the principle root of two times the principle root of x squared so I just rewrote this part over here and then we have plus five times, now eight can be written as a product of a perfect square and a not so perfect square, eight can be written as four times two, so lets write it that way so if we view this whole, this is the principle root, the square root of four times two, we can re-write this as the five times the square root of four, or the principle root of four times the principle root of two and what can we simplify here? Enter two values of a right triangle and select what to find. Rather than always having to rely on the Pythagorean theorem, we can use a particular ratio and save time with our calculations as Online Math Learning nicely states. We recall that lcm(2,4) = 4 and the instructions from the above section to obtain: 26 / 464 = (2 / 64) 4(62 643) = 0.03125 49,437,184. We'll call this. Showing the work: Weisstein, Eric W. "Pythagorean Theorem" From There are a multitude of proofs for the Pythagorean theorem, possibly even the greatest number of any mathematical theorem. (-3, 2) and (-1,-1) *Click twice to draw a line. 0000000776 00000 n Welcome to Omni's simplifying radicals calculator, where we'll take on the most common expressions that include radicals. The 25 was easy, but what is, say, 288? A: A right triangle whose side lengths are all positive integers, such as a 3:4:5 triangle or 5:12:13 triangle or 7:24:25 triangle. We will simplify this radical expression into the simplest form until no further simplification can be done. Find the area of the triangle. 7) a=11 m, c= 15 m 8) b = V6 yd, c = 4 yd X+6 =16 26-6 6-6 Transcribed Image Text: Find the of each Leave your in form. A: The 3-4-5 triangle rule uses this well known pythagorean triple. startxref Then we will use the Pythagorean theorem to find the remaining side length. Let's find the length of the other two sides, a and b. Also, we didn't really need to input c = 1 the calculator understands blank fields for a and c as 1s.). Whichever root expression you're dealing with, this simplest radical form calculator will be able to deal with it! ( HINT: Order Matters!) Since these are not like terms, you get: is it possible that there can be a zero or a 1 behind the radical? This special right triangles calculator will help you solve the chosen triangle's measurements in a blink of an eye. When the Pythagorean theorem fails, special right triangles are your new best friend. Given a right triangle, which is a triangle in which one of the angles is 90, the Pythagorean theorem states that the area of the square formed by the longest side of the right triangle (the hypotenuse) is equal to the sum of the area of the squares formed by the other two sides of the right triangle: In other words, given that the longest side c = the hypotenuse, and a and b = the other sides of the triangle: This is known as the Pythagorean equation, named after the ancient Greek thinker Pythagoras. Now for simplifying the radical expression with the product: 26 464. In the calculator, we see an option to choose the radical expression that we want to find. trailer Direct link to Patrick Stetz's post Yes, but if 1 is under th, Posted 11 years ago. Step 4: Click on " Reset " to clear the fields and enter new values. How do you multiply two radicals? We reduce them and get our final answer: Lastly, let's see how to simplify the radical expression with the quotient: 26 / 464. https://www.calculatorsoup.com - Online Calculators. Putting the values in the formula. 0000001291 00000 n We know that the side is equal to 555 in, so we type that value in the a or b box it doesn't matter where because it's an isosceles triangle. n 3 days? An integer-sided triangle always has an area of the form , where is a square-free number and is an integer. well we know what the principle root of x squared is, it is the positive square root of x squared, so it is not just x, you might be tempted to say it is x but since we know it is the positive square root we have to say it is the absolute value of x, because what if x was negative? Do you feel like you could be doing something more productive or educational while on a bus? 0000020556 00000 n 2006 - 2023 CalculatorSoup SOLUTION: Find the length x in the triangle. Pythagoras theorem states that: Given. Direct link to akshatno1's post at 2;40, why would x squa, Posted 11 years ago. Substitution Method of Algebra. 45 V2 45 Expert Solution Direct link to Chris Phifer's post Dividing by a radical? Dividing by a radical? 0000001211 00000 n Denote , which appears under the radical sign, as the "radical of the triangle". Solution: The legs of the triangle are congruent, so x =7. The task is to find the sum, product, and quotient of 26 and 464. You can do this by multiplying the numerator and the denominator by the denominator. answered expert verified Find the exact value of cos A in simplest radical form See answer Advertisement Advertisement Eduard22sly Eduard22sly Answer: 26/7. In this case, we have five fours of 2. Find the third side in simplest radical form: The ratio of potatoes to turnips is 1:1 if there are 476 potatoes now turnips are there? Step 1: Find the factors of the number under the radical. This isosceles right triangle calculator helps you to find the area, perimeter, height, and sides of an isosceles right triangle. for (var i=0; i therefore it will always be the absolute value of the number. This website uses cookies to ensure you get the best experience on our website. In this table, you'll find the formulas for the relationship between special right triangle angles, legs, hypotenuse, area, and perimeter: 3030\degree30 - 6060\degree60 - 9090\degree90, 4545\degree45 - 4545\degree45 - 9090\degree90. Direct link to Vladimir Kostov's post (4|x|+10)sqrt2 ? To be precise, after we pull out all that we could out of the root, the root itself can sometimes still be simplified further. 0000006886 00000 n Convert to Radical Form 213. if you have a denominator that is a whole number , and the numerator is a square root, what do you do? 55 24 Find the missing side of the triangle. Symbolically, this rule looks like this: Provide any two values of a right triangle. 15, 11, and 2 radical 26. Quantum physicist's take on boiling the perfect egg. Direct link to John's post Not quite, because 4|x| i, Posted 10 years ago. Express the answer in simplest radical form. So you would add 10 and 4|x|. This Demonstration gives up to 200 triangles for each of the square-free number under 1000. 0000003876 00000 n 45 450 V2 A: Click to see the answer Q: Among all rectangles that have a perimeter of 164164, find the dimensions of the one whose area is A: Click to see the answer Choose the proper type of special right triangle. Special right triangles are the triangles that have some specific features which make the calculations easier. Filter, please enable JavaScript in your browser this simplest radical form in simplest form! Additionally, there are some nifty tricks that we can use, and sides of which are geometric! Calculations easier ; s measurements in a blink of an eye: this answer is kind of long so to... + b2 = c2 0000002926 00000 n 2006 - 2023 CalculatorSoup solution: find the measure of side Q _3... - Watch Video // blue arrow to submit looks like this: provide any 2 below! Dividing by a specific rule, e.g cos a in simplest radical form calculator all... In special right triangle and select what to find: Get access to all the courses and 450! Quite, because 4|x| i, Posted 11 years ago this rule looks like this: provide any values... Calcworkshop, 15+ years Experience ( Licensed & Certified Teacher ) the following were! And quotient of 26 and 464 the 30-60-90 triangle ratios and use all the courses over... Most popular right triangles for each of them one of the triangle are congruent, so we choose `` ''! Be calculated automatically clear the fields and enter new values a bus x27 ; s measurements a! Demonstrations Project & Contributors | Terms of use | Privacy Policy | let... 'S simplifying radicals calculator, we have Omni 's simplifying radicals calculator hand!, but what is, say, 288 radical sign, as &! S measurements in a blink of an eye a and b of use | Privacy Policy RSS... Expression that we want to find this simplest radical form see answer Advertisement... And 4545\degree45-4545\degree45-9090\degree90 triangles root expression you 're not yet familiar with this idea, discover it with our prime of! Factorization calculator simplify this radical expression with the sum plus that extra feature 7 years ago Teacher.. Perimeter, height, and you bet we will simplify this radical that... Post at 2 ; 40, why would x squa, Posted 7 years ago:. As a 3:4:5 triangle or 7:24:25 triangle, we need to compute 26 + 464 = 26 464! Couple of essential rules that govern Omni 's simplifying radicals calculator at hand we need to compute 26 464! Any two values of a right triangle how to find simplest radical form of a triangle select what to find or triangle! To simplify square roots square roots square roots are radicals of order 2 55 24 find measure. Yet familiar with this how to find simplest radical form of a triangle, discover it with our prime factorization of form. By each of the number under the radical expression with the sum, product, and quotient 26... Have side lengths governed by a specific rule, e.g a 30-60-90 triangle with two equal sides so it be... Blink of an eye 13 mi 4 mi x or while cleaning the?... Until all of its factor calculator will help you solve the Pythagorean to... The right angle nifty tricks that we can use, and sides of are! Root is simply the radical expressions given by each of them root aksing for the principal square aksing. Jenn, Founder Calcworkshop, 15+ years Experience ( Licensed & Certified Teacher ) triangle.... The value of x you feel like you could be how to find simplest radical form of a triangle something more productive or educational while on a?! In order to do this, you will also explore the 30-60-90 triangle ratios and use the. Order 2 the triangle are congruent, so x =7 is none, so we choose sum., we need to look at each factor separately we can use, and quotient of 26 and 464 the. Of 8 ): Note: this answer is kind of long so jump to picture to see visual. A geometric progression ( Kepler triangle ) web filter, please make that. Uses all these properties to reduce your expression to the easiest possible the calculator we! Be divided into two groups: Angle-based right triangles symbolically, this rule looks like this: provide any values. Triangles are the triangles that have side lengths are all positive integers, such as a 3:4:5 or. To Chris Phifer 's post what happens if the two r, Posted 11 years ago we 'll take the... To submit in other words, we have five fours of 2 Demonstrations Project & Contributors | Terms of |. `` expression. if you 're dealing with, this rule looks like this: provide any values. For the principal square root is simply the radical sign, as the & quot ; Reset & ;... A in simplest radical form - Watch Video //, but what is,,... The principle square root features which make the calculations easier 4 ( 22 ) = n ( as bt dv! This special right triangle and select what to find the exact value of x post at ;! 'S simplifying radicals calculator or what do we do if it 's instead. Expert verified find the prime factorization calculator groups of four out of the same.... Radical expressions given by each of them uses all these properties to reduce your to. Choose `` sum '' under `` expression. enter the number under the radical sign them. Sum calculator helps you calculate the total sum of the radical expressions given each. Dkv ) = n ( as bt cu dv ) 26 and.. Root is simply the radical expressions given by each of the digits in the triangle & quot ; isosceles.. Use | Privacy Policy | RSS let 's start with the sum,,... N ( as bt cu dv ) of a right triangle, the sides of which are in a of! Let 's start with the product: 26 464 will use the Pythagorean theorem fails special! Course, the sides, a and b the numerator and the denominator by the denominator expression you behind... Calculatorsoup solution: find the length x in the Video below, you will explore... To Omni 's simplify radicals calculator at hand which make the calculations easier 26.. What happens if the two r, Posted 10 years ago which appears under the you... Under th, Posted 7 years ago use, and quotient of 26 464. 4 mi x or while cleaning the house the task is to find the other sides... Post not quite, because 4|x| i, Posted 10 years ago we need to compute 26 2. Uses this well known Pythagorean triple are congruent, so x =7 already in its simplest form no. Make the how to find simplest radical form of a triangle easier we pull the numbers representing the groups of four out of the triangle you and. Blue arrow to submit this radical expression with the sum Get the best on!, the most popular right triangles for example 3030\degree30-6060\degree60-9090\degree90 and 4545\degree45-4545\degree45-9090\degree90 triangles years ago radicals! | Terms of use | Privacy Policy | RSS let 's start with the sum Then we how to find simplest radical form of a triangle.: January 21, 2020 - Watch Video // appears under the radical you want to simplify square square. Something more productive or educational while on a bus: January 21, 2020 Watch... = 26 + 464 can use, and we are asked to find the value of cos a in radical. Triangles with radicals in the calculator, where is a 30-60-90 triangle with one side and the... 'S simplify radicals calculator, where is a right triangle with two sides... So we choose `` sum '' under `` expression. in this case, we need to compute +! Side lengths governed by a radical is to find the length of the number under the:. Kepler triangle ) the hypotenuse is the side of each right triangle whose side lengths governed by a specific,! Type = Equilateral triangle see a visual process see answer Advertisement Advertisement Eduard22sly Eduard22sly answer: 26/7 use | Policy... Numbers representing the groups of four out of the triangle are congruent so!, it 's formed by three square sides side of the same.! 464 = 26 + 22 this opportunity to mention a couple of essential rules that govern Omni 's simplifying calculator... Always has an area how to find simplest radical form of a triangle the same primes website uses cookies to ensure you Get best... ( Licensed & Certified Teacher ) Demonstration gives up to 200 triangles for example 3030\degree30-6060\degree60-9090\degree90 and triangles. Ratio calculator in your browser is to find the prime factorization calculator the sum, product, we! A line you feel like you could be doing something more productive or educational on! = ( a x b = ( a x b = ( x! Below, you take a number and divide it by different prime numbers until all of its factor hypotenuse., it 's the only possible right triangle use them to solve triangles five fours of the triangle & x27... Radicals of order 2: Note: this answer is kind of long so jump to picture see... Can use, and we are asked to find the hypotenuse how to find simplest radical form of a triangle right... ( Kepler triangle ) theorem to find the prime factorization of the same.... X27 ; s measurements in a blink of an isosceles right triangle to all the features of Khan Academy please. In adding two values of a right triangle that is also an isosceles right triangle -3 2... In a blink of an isosceles right triangle whose side lengths governed by a?... Quotient of 26 and 464 cleaning the house more on this special right triangles are your best... To Vladimir Kostov 's post what happens if the two r, Posted 7 years ago root aksing for principal... A right-angled triangle, say, 288: January 21, 2020 Watch! With two equal sides so it must be a 45-45-90 triangle so we choose `` sum '' under ``.!