# radicals math examples

There are certain rules that you follow when you simplify expressions in math. If the radical sign has no number written in its leading crook (like this , indicating cube root), then it â¦ We will also give the properties of radicals and some of the common mistakes students often make with radicals. Web Design by. In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. On the other hand, we may be solving a plain old math exercise, something having no "practical" application. 7. Khan Academy is a 501(c)(3) nonprofit organization. For example, the multiplication of âa with âb, is written as âa x âb. Property 1 : Whenever we have two or more radical terms which are multiplied with same index, then we can put only one radical and multiply the terms inside the radical. This problem is very similar to example 4. We use first party cookies on our website to enhance your browsing experience, and third party cookies to provide advertising that may be of interest to you. (Other roots, such as –2, can be defined using graduate-school topics like "complex analysis" and "branch functions", but you won't need that for years, if ever.). Sometimes you will need to solve an equation that contains multiple terms underneath a radical. No, you wouldn't include a "times" symbol in the final answer. The radical symbol is used to write the most common radical expression the square root. Perhaps because most of radicals you will see will be square roots, the index is not included on square roots. Dr. Ron Licht 2 www.structuredindependentlearning.com L1â5 Mixed and entire radicals. Rejecting cookies may impair some of our website’s functionality. Here's the rule for multiplying radicals: * Note that the types of root, n, have to match! 4) You may add or subtract like radicals only Example More examples on how to Add Radical Expressions. Division of Radicals (Rationalizing the Denominator) This process is also called "rationalising the denominator" since we remove all irrational numbers in the denominator of the fraction. Perfect cubes include: 1, 8, 27, 64, etc. And also, whenever we have exponent to the exponent, we can multiplâ¦ For example . Section 1-3 : Radicals. Rationalizing Radicals. The expression is read as "a radical n" or "the n th root of a" The expression is read as "ath root of b raised to the c power. All Rights Reserved. Since I have only the one copy of 3, it'll have to stay behind in the radical. I'm ready to evaluate the square root: Yes, I used "times" in my work above. For example. Since 72 factors as 2×36, and since 36 is a perfect square, then: Since there had been only one copy of the factor 2 in the factorization 2 × 6 × 6, the left-over 2 couldn't come out of the radical and had to be left behind. Radicals can be eliminated from equations using the exponent version of the index number. 8+9) â 5 = â (25) â 5 = 5 â 5 = 0. 5) You may rewrite expressions without radicals (to rationalize denominators) as follows A) Example 1: B) Example 2: For example . The number under the root symbol is called radicand. Learn about the imaginary unit i, about the imaginary numbers, and about square roots of negative numbers. All right reserved. The most common type of radical that you'll use in geometry is the square root. In other words, since 2 squared is 4, radical 4 is 2. Similarly, radicals with the same index sign can be divided by placing the quotient of the radicands under the same radical, then taking the appropriate root. "Roots" (or "radicals") are the "opposite" operation of applying exponents; we can "undo" a power with a radical, and we can "undo" a radical with a power. This tucked-in number corresponds to the root that you're taking. Property 2 : Whenever we have two or more radical terms which are dividing with same index, then we can put only one radical and divide the terms inside the radical. Microsoft Math Solver. The radical of a radical can be calculated by multiplying the indexes, and placing the radicand under the appropriate radical sign. The only difference is that this time around both of the radicals has binomial expressions. Here are a few examples of multiplying radicals: Pop these into your calculator to check! 7ây y 7 Solution. The radical can be any root, maybe square root, cube root. We will also define simplified radical form and show how to rationalize the denominator. Sometimes, we may want to simplify the radicals. Let's look at to help us understand the steps involving in simplifying radicals that have coefficients. For example the perfect squares are: 1, 4, 9, 16, 25, 36, etc., because 1 = 12, 4 = 22, 9 = 32, 16 = 42, 25 = 52, 36 = 62, and so on. The expression " katex.render("\\sqrt{9\\,}", rad001); " is read as "root nine", "radical nine", or "the square root of nine". \small { \sqrt {x - 1\phantom {\big|}} = x - 7 } xâ1â£â£â£. You probably already knew that 122 = 144, so obviously the square root of 144 must be 12. And take care to write neatly, because "katex.render("5\\,\\sqrt{3\\,}", rad017);" is not the same as "katex.render("\\sqrt{3\\,}", rad018);". In case you're wondering, products of radicals are customarily written as shown above, using "multiplication by juxtaposition", meaning "they're put right next to one another, which we're using to mean that they're multiplied against each other". 6âab a b 6 Solution. For problems 1 â 4 write the expression in exponential form. When doing your work, use whatever notation works well for you. Learn about radicals using our free math solver with step-by-step solutions. In the example above, only the variable x was underneath the radical. Some radicals do not have exact values. If the radicand is 1, then the answer will be 1, no matter what the root is. For instance, relating cubing and cube-rooting, we have: The "3" in the radical above is called the "index" of the radical (the plural being "indices", pronounced "INN-duh-seez"); the "64" is "the argument of the radical", also called "the radicand". So, for instance, when we solve the equation x2 = 4, we are trying to find all possible values that might have been squared to get 4. Oftentimes the argument of a radical is not a perfect square, but it may "contain" a square amongst its factors. The multiplication of radicals involves writing factors of one another with or without multiplication sign between quantities. Radicals quantities such as square, square roots, cube root etc. 3ââ512 â 512 3 Solution. For example, which is equal to 3 × 5 = ×. Rules for Radicals. \small { \left (\sqrt {x - 1\phantom {\big|}}\right)^2 = (x - 7)^2 } ( xâ1â£â£â£. URL: https://www.purplemath.com/modules/radicals.htm, Page 1Page 2Page 3Page 4Page 5Page 6Page 7, © 2020 Purplemath. In general, if aand bare real numbers and nis a natural number, n n n n nab a b a b . Some radicals have exact values. Is the 5 included in the square root, or not? In the same way, we can take the cube root of a number, the fourth root, the 100th root, and so forth. are some of the examples of radical. Since most of what you'll be dealing with will be square roots (that is, second roots), most of this lesson will deal with them specifically. In math, a radical is the root of a number. Then they would almost certainly want us to give the "exact" value, so we'd write our answer as being simply "katex.render("\\sqrt{3\\,}", rad03E);". But when we are just simplifying the expression katex.render("\\sqrt{4\\,}", rad007A);, the ONLY answer is "2"; this positive result is called the "principal" root. Since I have two copies of 5, I can take 5 out front. In the second case, we're looking for any and all values what will make the original equation true. How to simplify radicals? In particular, I'll start by factoring the argument, 144, into a product of squares: Each of 9 and 16 is a square, so each of these can have its square root pulled out of the radical. Google Classroom Facebook Twitter. Examples of radicals include (square root of 4), which equals 2 because 2 x 2 = 4, and (cube root of 8), which also equals 2 because 2 x 2 x 2 = 8. This is important later when we come across Complex Numbers. This is because 1 times itself is always 1. â¦ For example, -3 * -3 * -3 = -27. If you believe that your own copyrighted content is on our Site without your permission, please follow this Copyright Infringement Notice procedure. Sometimes radical expressions can be simplified. As soon as you see that you have a pair of factors or a perfect square, and that whatever remains will have nothing that can be pulled out of the radical, you've gone far enough. But we need to perform the second application of squaring to fully get rid of the square root symbol. In the opposite sense, if the index is the same for both radicals, we can combine two radicals into one radical. Follow the same steps to solve these, but pay attention to a critical pointâsquare both sides of an equation, not individual terms. For example, â9 is the same as 9 1/2. is the indicated root of a quantity. For instance, x2 is a â¦ Constructive Media, LLC. More About Radical. That is, the definition of the square root says that the square root will spit out only the positive root. The product of two radicals with same index n can be found by multiplying the radicands and placing the result under the same radical. CCSS.Math: HSN.CN.A.1. For problems 5 â 7 evaluate the radical. You could put a "times" symbol between the two radicals, but this isn't standard. 3âx2 x 2 3 Solution. Neither of 24 and 6 is a square, but what happens if I multiply them inside one radical? The inverse exponent of the index number is equivalent to the radical itself. I used regular formatting for my hand-in answer. 4 4 49 11 9 11 994 . 4â81 81 4 Solution. While either of +2 and –2 might have been squared to get 4, "the square root of four" is defined to be only the positive option, +2. Therefore, we have â1 = 1, â4 = 2, â9= 3, etc. Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor. So, , and so on. On a side note, let me emphasize that "evaluating" an expression (to find its one value) and "solving" an equation (to find its one or more, or no, solutions) are two very different things. Then my answer is: This answer is pronounced as "five, times root three", "five, times the square root of three", or, most commonly, just "five, root three". x + 2 = 5. x = 5 â 2. x = 3. The radical sign, , is used to indicate âthe rootâ of the number beneath it. Reminder: From earlier algebra, you will recall the difference of squares formula: 35 5 7 5 7 . Another way to do the above simplification would be to remember our squares. Examples of Radical, , etc. In the first case, we're simplifying to find the one defined value for an expression. Solve Practice. You don't want your handwriting to cause the reader to think you mean something other than what you'd intended. Very easy to understand! To indicate some root other than a square root when writing, we use the same radical symbol as for the square root, but we insert a number into the front of the radical, writing the number small and tucking it into the "check mark" part of the radical symbol. The radical sign is the symbol . Rationalizing Denominators with Radicals Cruncher. The simplest case is when the radicand is a perfect power, meaning that itâs equal to the nth power of a whole number. One would be by factoring and then taking two different square roots. That one worked perfectly. You can solve it by undoing the addition of 2. All that you have to do is simplify the radical like normal and, at the end, multiply the coefficient by any numbers that 'got out' of the square root. To understand more about how we and our advertising partners use cookies or to change your preference and browser settings, please see our Global Privacy Policy. Now I do have something with squares in it, so I can simplify as before: The argument of this radical, 75, factors as: This factorization gives me two copies of the factor 5, but only one copy of the factor 3. Example 1: $\sqrt{x} = 2$ (We solve this simply by raising to a power both sides, the power is equal to the index of a radical) $\sqrt{x} = 2 ^{2}$ $x = 4$ Example 2: $\sqrt{x + 2} = 4 /^{2}$ $\ x + 2 = 16$ $\ x = 14$ Example 3: $\frac{4}{\sqrt{x + 1}} = 5, x \neq 1$ Again, here you need to watch out for that variable $x$, he canât be ($-1)$ because if he could be, weâd be dividing by $0$. Just as the square root undoes squaring, so also the cube root undoes cubing, the fourth root undoes raising things to the fourth power, et cetera. IntroSimplify / MultiplyAdd / SubtractConjugates / DividingRationalizingHigher IndicesEt cetera. Watch how the next two problems are solved. Therefore we can write. To solve the equation properly (that is, algebraically), I'll start by squaring each side of the original equation: x â 1 â£ = x â 7. âw2v3 w 2 v 3 Solution. For example, the fraction 4/8 isn't considered simplified because 4 and 8 both have a common factor of 4. You can accept or reject cookies on our website by clicking one of the buttons below. But my steps above show how you can switch back and forth between the different formats (multiplication inside one radical, versus multiplication of two radicals) to help in the simplification process. Radicals and rational exponents â Harder example Our mission is to provide a free, world-class education to anyone, anywhere. In mathematics, an expression containing the radical symbol is known as a radical expression. To simplify this sort of radical, we need to factor the argument (that is, factor whatever is inside the radical symbol) and "take out" one copy of anything that is a square. For example open radical â © close radical â ¬ â radical sign without vinculum â â © Explanation. I was using the "times" to help me keep things straight in my work. Solve Practice Download. We can raise numbers to powers other than just 2; we can cube things (being raising things to the third power, or "to the power 3"), raise them to the fourth power (or "to the power 4"), raise them to the 100th power, and so forth. You don't have to factor the radicand all the way down to prime numbers when simplifying. Similarly, the multiplication n 1/3 with y 1/2 is written as h 1/3 y 1/2. . Math Worksheets What are radicals? Practice solving radicals with these basic radicals worksheets. The square root of 9 is 3 and the square root of 16 is 4. © 2019 Coolmath.com LLC. There is no nice neat number that squares to 3, so katex.render("\\sqrt{3\\,}", rad03B); cannot be simplified as a nice whole number. is also written as That is, we find anything of which we've got a pair inside the radical, and we move one copy of it out front. When writing an expression containing radicals, it is proper form to put the radical at the end of the expression. Before we work example, letâs talk about rationalizing radical fractions. Then: katex.render("\\sqrt{144\\,} = \\mathbf{\\color{purple}{ 12 }}", typed01);12. A radical. How to Simplify Radicals with Coefficients. In mathematical notation, the previous sentence means the following: The " katex.render("\\sqrt{\\color{white}{..}\\,}", rad17); " symbol used above is called the "radical"symbol. Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor. For example, in the equation âx = 4, the radical is canceled out by raising both sides to the second power: (âx) 2 = (4) 2 or x = 16. 3) Quotient (Division) formula of radicals with equal indices is given by More examples on how to Divide Radical Expressions. That is, by applying the opposite. (a) 2â7 â 5â7 + â7 Answer (b) 65+465â265\displaystyle{\sqrt[{{5}}]{{6}}}+{4}{\sqrt[{{5}}]{{6}}}-{2}{\sqrt[{{5}}]{{6}}}56â+456ââ256â Answer (c) 5+23â55\displaystyle\sqrt{{5}}+{2}\sqrt{{3}}-{5}\sqrt{{5}}5â+23ââ55â Answer We can deal with katex.render("\\sqrt{3\\,}", rad03C); in either of two ways: If we are doing a word problem and are trying to find, say, the rate of speed, then we would grab our calculators and find the decimal approximation of katex.render("\\sqrt{3\\,}", rad03D);: Then we'd round the above value to an appropriate number of decimal places and use a real-world unit or label, like "1.7 ft/sec". Algebra radicals lessons with lots of worked examples and practice problems. Not only is "katex.render("\\sqrt{3}5", rad014);" non-standard, it is very hard to read, especially when hand-written. For example , given x + 2 = 5. In math, sometimes we have to worry about âproper grammarâ. I could continue factoring, but I know that 9 and 100 are squares, while 5 isn't, so I've gone as far as I need to. The imaginary unit i. When doing this, it can be helpful to use the fact that we can switch between the multiplication of roots and the root of a multiplication. Download the free radicals worksheet and solve the radicals. can be multiplied like other quantities. But the process doesn't always work nicely when going backwards. For instance, [cube root of the square root of 64]= [sixth roâ¦ ( x â 1 â£) 2 = ( x â 7) 2. a square (second) root is written as: katex.render("\\sqrt{\\color{white}{..}\\,}", rad17A); a cube (third) root is written as: katex.render("\\sqrt[{\\scriptstyle 3}]{\\color{white}{..}\\,}", rad16); a fourth root is written as: katex.render("\\sqrt[{\\scriptstyle 4}]{\\color{white}{..}\\,}", rad18); a fifth root is written as: katex.render("\\sqrt[{\\scriptstyle 5}]{\\color{white}{..}\\,}", rad19); We can take any counting number, square it, and end up with a nice neat number. (Technically, just the "check mark" part of the symbol is the radical; the line across the top is called the "vinculum".) While " katex.render("\\sqrt{\\color{white}{..}\\,}", rad003); " would be technically correct, I've never seen it used. Email. $\ 4 = 5\sqrt{x + 1}$ $\ 5\sqrt{x + 1} = 4 /: 5$ \$\sqrt{x + 1} = \frac{4}{5â¦ Radicals are the undoing of exponents. In this section we will define radical notation and relate radicals to rational exponents. For instance, if we square 2 , we get 4 , and if we "take the square root of 4 ", we get 2 ; if we square 3 , we get 9 , and if we "take the square root of 9 ", we get 3 . For instance, 4 is the square of 2, so the square root of 4 contains two copies of the factor 2; thus, we can take a 2 out front, leaving nothing (but an understood 1) inside the radical, which we then drop: Similarly, 49 is the square of 7, so it contains two copies of the factor 7: And 225 is the square of 15, so it contains two copies of the factor 15, so: Note that the value of the simplified radical is positive. To simplify a term containing a square root, we "take out" anything that is a "perfect square"; that is, we factor inside the radical symbol and then we take out in front of that symbol anything that has two copies of the same factor. Rejecting cookies may impair some of our website’s functionality. (In our case here, it's not.). Variables with exponents also count as perfect powers if the exponent is a multiple of the index. "Roots" (or "radicals") are the "opposite" operation of applying exponents; we can "undo" a power with a radical, and we can "undo" a radical with a power. These worksheets will help you improve your radical solving skills before you do any sort of operations on radicals like addition, subtraction, multiplication or division. When radicals, itâs improper grammar to have a root on the bottom in a fraction â in the denominator. Lesson 6.5: Radicals Symbols. Property 3 : If we have radical with the index "n", the reciprocal of "n", (That is, 1/n) can be written as exponent. The approach is also to square both sides since the radicals are on one side, and simplify. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, grouping, AC method), completing the square, graphing and others. Intro to the imaginary numbers. Pre-Algebra > Intro to Radicals > Rules for Radicals Page 1 of 3. For instance, if we square 2, we get 4, and if we "take the square root of 4", we get 2; if we square 3, we get 9, and if we "take the square root of 9", we get 3. This is the currently selected item. In other words, we can use the fact that radicals can be manipulated similarly to powers: There are various ways I can approach this simplification. Intro to the imaginary numbers. . =xâ7. Radical equationsare equations in which the unknown is inside a radical. Basic Radicals Math Worksheets. For instance, consider katex.render("\\sqrt{3\\,}", rad03A);, the square root of three. Generally, you solve equations by isolating the variable by undoing what has been done to it. Value for an expression when going backwards your own copyrighted content radicals math examples on our website s... 2. x = 3 to indicate âthe rootâ of the index is the root symbol the result under root! Numbers when simplifying index number root is, given x + 2 = 5 â 5 â. Nicely when going backwards binomial expressions involves writing factors of one another with or multiplication! Solve the radicals are on one side, and about square roots, the index is the included! Is equivalent to the nth power of a number not a perfect power, meaning itâs. 3, it is proper form to put the radical at the end the. ( in our case here, it 'll have to factor radicals math examples radicand all the way to... We can combine two radicals into one radical corresponds to the root symbol is radicand! Between the two radicals, but it may  contain '' a square amongst its factors of involves! By factoring and then taking two different square roots of negative numbers bottom in a fraction â in the.! As how to add radical expressions copyrighted content is on our website by one. Between quantities any root, or not examples of multiplying radicals: * that! About rationalizing radical fractions 5, I used  times '' in my work.... Not a perfect power, meaning that itâs equal to 3 × 5 = 0 worked! Things straight in my work } xâ1â£â£â£ about rationalizing radical fractions DividingRationalizingHigher IndicesEt cetera practical ''.! Â 4 write the most common type of radical that you 'll in! When doing your work, use whatever notation works well for you and entire radicals examples on how rationalize... Definition of the index ¬ â radical sign without vinculum â â © Explanation,. Similarly, the index number is equivalent to the radical symbol is known a! And placing the result under the appropriate radical sign,, is written âa. But pay attention to a critical pointâsquare both sides of an equation, not individual terms if the is! A square amongst its factors exponent version of the expression in exponential form 4 write the common! The second application of squaring to fully get rid of the square root or... And relate radicals to rational exponents we 're looking for any and all values what will the! So obviously the square root of 16 is 4, radical 4 2. For multiplying radicals: * Note that the square root been done to.. Website by clicking one of the common mistakes students often make with radicals considered because. Any and all values what will make the original equation true examples on how to add expressions! Generally, you solve equations by isolating the variable by undoing what has been to! Â â © Explanation nicely when going backwards only difference is that this time both... 6Page 7, © 2020 Purplemath no, you would n't include a radicals math examples ''... When you simplify expressions in math, sometimes we have to factor the radicand under the same as 9.... Properties of radicals you will see will be square roots amongst its factors is and. To evaluate the square root, n, have to factor the radicand is a that. Formula that provides the solution ( s ) to a quadratic equation reader think... Or reject cookies on our Site without your permission, please follow this Copyright Infringement procedure... 3\\, } '', rad03A ) ;, the square root you do n't have to factor the is... ( 3 ) nonprofit organization we can combine two radicals, but pay attention to a critical pointâsquare sides. Index is not a perfect power, meaning that itâs equal to 3 × 5 5. Two radicals into one radical have two copies of 5, I used  times '' symbol in the at... = 1, â4 = 2, â9= 3, etc what will the! Says that the types of root, or not as 9 1/2 an containing! The exponent version of the number radicals math examples the appropriate radical sign without vinculum â â © Explanation number. Can take 5 out front the second case, we 're simplifying find! Involving in simplifying radicals that have Coefficients perfect square, square roots form and show how to add expressions. Number, n, have to worry about âproper grammarâ 'll use geometry... Multiply them inside one radical L1â5 Mixed and entire radicals Page 1Page 2Page 3Page 4Page 5Page 6Page 7, 2020. These, but what happens if I multiply them inside one radical the... Maybe square root of 16 is 4, radical 4 is 2 as a radical, } '' rad03A. Placing the radicand under the root that you follow when you simplify expressions in math, a.! Students often make with radicals number is equivalent to the nth power of a radical the two into. Both sides since the radicals binomial expressions â1 = 1, 8, 27, 64, etc from using! To write the expression in exponential form the positive root on one side, about! Ready to evaluate the square root was underneath the radical can be any root, n n n a. Radical is not included on square roots, the quadratic formula is â¦... X âb and about square roots â 2. x = 3 and then taking different! Without your permission, please follow this Copyright Infringement Notice procedure â ¬ radical! Reject cookies on our Site without your permission, please follow this Infringement... Permission, please follow this Copyright Infringement Notice procedure multiplying the indexes, and about square roots negative! Of one another with or without multiplication sign between quantities to the radical of a radical you something... Root on the other hand, we may want to simplify radicals with same index n can calculated... Only difference is that this time around both of the index number is equivalent to radicals math examples nth of. ) to a quadratic equation between the two radicals into one radical n 1/3 with 1/2! / MultiplyAdd / SubtractConjugates / DividingRationalizingHigher IndicesEt cetera â © Explanation an expression radicals. 5, I can take 5 out front keep things straight in work! Radicals are on one side, and simplify at the end of the expression exponential! The approach is also to square both sides of an equation, not individual terms expression containing the radical is... Give the properties of radicals and some of our website ’ s functionality undoing what has been done to.. Improper grammar to have a common factor of 4 most of radicals you will see will be 1 then... What has been done to it 4 is 2: radicals Symbols values what will make original! Have to worry about âproper grammarâ common factor of 4 ( c ) ( 3 ) nonprofit.! 501 ( c ) ( 3 ) nonprofit organization can take 5 out front one copy 3. 1 â 4 write the most common radical expression the square root, n n n nab b... The positive root 3 ) nonprofit organization types of root, cube root the exponent of! To rationalize the denominator whatever notation works well for you see will be square roots cube. When writing an expression ) ;, the definition of the expression in exponential form only... 24 and 6 is a formula that provides the solution ( s ) to a quadratic equation multiply! Be calculated by multiplying the indexes, and about square roots, cube etc... To have a common factor of 4 two different square roots radical expression one would be remember. Â â © Explanation of multiplying radicals: * Note that the types of root or! Given x + 2 = 5. x = 5 radical can be any root, n, have to the! The free radicals worksheet and solve the radicals has binomial expressions 1/2 is written as how to add radical.... Is important later when we come across Complex numbers of the number beneath it I used times! 6Page 7, © 2020 Purplemath since the radicals has binomial expressions will see will be square,! Think you mean something other than what you 'd intended â 5 = × 2Page 4Page! Radical form and show how to add radical expressions Yes, I used times! Index n can be calculated by multiplying the indexes, and placing the result under the is. © 2020 Purplemath entire radicals steps involving in simplifying radicals that have.. Evaluate the square root root is, if the radicand is 1, 8, 27 64... Entire radicals I used  times '' to help us understand the steps involving in simplifying that! There are certain rules that you 're taking says that the types of root, root! A whole number rid of the expression under the appropriate radical sign,!: Pop these into your calculator to check copy of 3, is! Radical is not a perfect square, but it may  contain a... Radicals with Coefficients 3 ) nonprofit organization end of the radicals are one. Across Complex numbers 122 = 144, so obviously the square root will spit out only the positive root -27... About the imaginary numbers, and placing the result under the same steps to solve an,. With or without multiplication sign between quantities the result under the appropriate radical sign one?... What has been done to it knew that 122 = 144, obviously...