Example 4: Simplify the radical expression \sqrt {48} . Repeat the process until such time when the radicand no longer has a perfect square factor. To simplify this radical number, try factoring it out such that one of the factors is a perfect square. Step 2. A spider connects from the top of the corner of cube to the opposite bottom corner. Example 12: Simplify the radical expression \sqrt {125} . Example 4 : Simplify the radical expression : √243 - 5√12 + √27. If you're seeing this message, it means we're having trouble loading external resources on our website. Find the value of a number n if the square root of the sum of the number with 12 is 5. How many zones can be put in one row of the playground without surpassing it? By multiplication, simplify both the expression inside and outside the radical to get the final answer as: To solve such a problem, first determine the prime factors of the number inside the radical. Simplify each of the following expression. You just need to make sure that you further simplify the leftover radicand (stuff inside the radical symbol). Calculate the speed of the wave when the depth is 1500 meters. Examples There are a couple different ways to simplify this radical. Simplify. Example 9: Simplify the radical expression \sqrt {400{h^3}{k^9}{m^7}{n^{13}}} . Next, express the radicand as products of square roots, and simplify. Therefore, we need two of a kind. Solution: a) 14x + 5x = (14 + 5)x = 19x b) 5y – 13y = (5 –13)y = –8y c) p – 3p = (1 – 3)p = – 2p. The following are the steps required for simplifying radicals: –3√(2 x 2 x 2 x2 x 3 x 3 x 3 x x 7 x y 5). The answer must be some number n found between 7 and 8. If we do have a radical sign, we have to rationalize the denominator. Step 1. Let’s explore some radical expressions now and see how to simplify them. Example 13: Simplify the radical expression \sqrt {80{x^3}y\,{z^5}}. 1 6. Roots and radical expressions 1. The idea of radicals can be attributed to exponentiation, or raising a number to a given power. Let’s find a perfect square factor for the radicand. Multiply by . Example 11: Simplify the radical expression \sqrt {32} . Here’s a radical expression that needs simplifying, . One way of simplifying radical expressions is to break down the expression into perfect squares multiplying each other. For this problem, we are going to solve it in two ways. • Multiply and divide rational expressions. Adding and Subtracting Radical Expressions, That’s the reason why we want to express them with even powers since. 7. The goal is to show that there is an easier way to approach it especially when the exponents of the variables are getting larger. Example 2: Simplify by multiplying. Determine the index of the radical. Actually, any of the three perfect square factors should work. You can do some trial and error to find a number when squared gives 60. Radicals, radicand, index, simplified form, like radicals, addition/subtraction of radicals. It is okay to multiply the numbers as long as they are both found under the radical … It’s okay if ever you start with the smaller perfect square factors. 4. 27. Thus, the answer is. Calculate the area of a right triangle which has a hypotenuse of length 100 cm and 6 cm width. A rectangular mat is 4 meters in length and √ (x + 2) meters in width. Here it is! Extract each group of variables from inside the radical, and these are: 2, 3, x, and y. Move only variables that make groups of 2 or 3 from inside to outside radicals. Let’s do that by going over concrete examples. If the area of the playground is 400, and is to be subdivided into four equal zones for different sporting activities. One method of simplifying this expression is to factor and pull out groups of a 3, as shown below in this example. Sometimes radical expressions can be simplified. Simplifying Radicals Operations with Radicals 2. • Add and subtract rational expressions. The main approach is to express each variable as a product of terms with even and odd exponents. Simplify. 2 2 2 2 2 2 1 1 2 4 3 9 4 16 5 25 6 36 = = = = = = 1 1 4 2 9 3 16 4 25 5 36 6 = = = = = = 2 2 2 2 2 2 7 49 8 64 9 81 10 100 11 121 12 144 = = = = = = 49 7 64 8 81 9 100 10 121 11 144 12 = = = = = = 3. By quick inspection, the number 4 is a perfect square that can divide 60. Find the largest perfect square that is a factor of the radicand (just like before) 4 is the largest perfect square that is a factor of 8. A school auditorium has 3136 total number of seats, if the number of seats in the row is equal to the number of seats in the columns. Thanks to all of you who support me on Patreon. 5. Example 2: Simplify the radical expression \sqrt {60}. This is an easy one! We need to recognize how a perfect square number or expression may look like. Mary bought a square painting of area 625 cm 2. Simplest form. Add and . For example ; Since the index is understood to be 2, a pair of 2s can move out, a pair of xs can move out and a pair of ys can move out. This is achieved by multiplying both the numerator and denominator by the radical in the denominator. 3. And it checks when solved in the calculator. since √x is a real number, x is positive and therefore |x| = x. is not a real number since -x 2 - 1 is always negative. 1. If the term has an even power already, then you have nothing to do. There should be no fraction in the radicand. 8. More so, the variable expressions above are also perfect squares because all variables have even exponents or powers. To simplify complicated radical expressions, we can use some definitions and rules from simplifying exponents. In this last video, we show more examples of simplifying a quotient with radicals. You could start by doing a factor tree and find all the prime factors. These properties can be used to simplify radical expressions. Examples C) If n is an ODD positive integer then Examples Questions With Answers Rewrite, if possible, the following expressions without radicals (simplify) Solutions to the Above Problems The index of the radical 3 is odd and equal to the power of the radicand. 4. Notice that the square root of each number above yields a whole number answer. Note, for each pair, only one shows on the outside. My apologies in advance, I kept saying rational when I meant to say radical. Radical expressions come in many forms, from simple and familiar, such as[latex] \sqrt{16}[/latex], to quite complicated, as in [latex] \sqrt[3]{250{{x}^{4}}y}[/latex]. Raise to the power of . What rule did I use to break them as a product of square roots? Divide the number by prime factors such as 2, 3, 5 until only left numbers are prime. Otherwise, you need to express it as some even power plus 1. So we expect that the square root of 60 must contain decimal values. Each side of a cube is 5 meters. Example 1: to simplify $(\sqrt{2}-1)(\sqrt{2}+1)$ type (r2 - 1)(r2 + 1). $1 per month helps!! What does this mean? 5. Algebra. Radical Expressions and Equations. Similar radicals. Simplify the following radicals. Multiply and . The paired prime numbers will get out of the square root symbol, while the single prime will stay inside. Example 5: Simplify the radical expression \sqrt {200} . In order to simplify radical expressions, you need to be aware of the following rules and properties of radicals 1) From definition of n th root(s) and principal root Examples ... More examples on how to Rationalize Denominators of Radical Expressions. Calculate the value of x if the perimeter is 24 meters. Fractional radicand . We can add or subtract radical expressions only when they have the same radicand and when they have the same radical type such as square roots. For example, in not in simplified form. Think of them as perfectly well-behaved numbers. A worked example of simplifying an expression that is a sum of several radicals. Let’s simplify this expression by first rewriting the odd exponents as powers of an even number plus 1. Find the prime factors of the number inside the radical. Example 1. Use the power rule to combine exponents. Multiplication of Radicals Simplifying Radical Expressions Example 3: \(\sqrt{3} \times \sqrt{5} = ?\) A. The number 16 is obviously a perfect square because I can find a whole number that when multiplied by itself gives the target number. In addition, those numbers are perfect squares because they all can be expressed as exponential numbers with even powers. 9. However, I hope you can see that by doing some rearrangement to the terms that it matches with our final answer. Example 4 – Simplify: Step 1: Find the prime factorization of the number inside the radical and factor each variable inside the radical. However, it is often possible to simplify radical expressions, and that may change the radicand. \(\sqrt{8}\) C. \(3\sqrt{5}\) D. \(5\sqrt{3}\) E. \(\sqrt{-1}\) Answer: The correct answer is A. Perfect Powers 1 Simplify any radical expressions that are perfect squares. Pairing Method: This is the usual way where we group the variables into two and then apply the square root operation to take the variable outside the radical symbol. Express the odd powers as even numbers plus 1 then apply the square root to simplify further. A rectangular mat is 4 meters in length and √(x + 2) meters in width. • Simplify complex rational expressions that involve sums or di ff erences … A radical can be defined as a symbol that indicate the root of a number. Algebra Examples. 2nd level. A kite is secured tied on a ground by a string. For example, the sum of \(\sqrt{2}\) and \(3\sqrt{2}\) is \(4\sqrt{2}\). For the number in the radicand, I see that 400 = 202. Square root, cube root, forth root are all radicals. You da real mvps! However, the key concept is there. If you have radical sign for the entire fraction, you have to take radical sign separately for numerator and denominator. W E SAY THAT A SQUARE ROOT RADICAL is simplified, or in its simplest form, when the radicand has no square factors.. A radical is also in simplest form when the radicand is not a fraction.. \(\sqrt{15}\) B. The calculator presents the answer a little bit different. An expression is considered simplified only if there is no radical sign in the denominator. Raise to the power of . Examples of How to Simplify Radical Expressions. 1. This type of radical is commonly known as the square root. Now for the variables, I need to break them up into pairs since the square root of any paired variable is just the variable itself. We hope that some of those pieces can be further simplified because the radicands (stuff inside the symbol) are perfect squares. In this example, we simplify √(2x²)+4√8+3√(2x²)+√8. Looks like the calculator agrees with our answer. Going through some of the squares of the natural numbers…. Calculate the amount of woods required to make the frame. Starting with a single radical expression, we want to break it down into pieces of “smaller” radical expressions. 2 1) a a= b) a2 ba= × 3) a b b a = 4. See below 2 examples of radical expressions. Let’s deal with them separately. Multiply the variables both outside and inside the radical. :) https://www.patreon.com/patrickjmt !! It must be 4 since (4) (4) = 4 2 = 16. The solution to this problem should look something like this…. Calculate the value of x if the perimeter is 24 meters. Or you could start looking at perfect square and see if you recognize any of them as factors. √22 2 2. If you have square root (√), you have to take one term out of the square root for every two same terms multiplied inside the radical. Great! Our equation which should be solved now is: Subtract 12 from both side of the expression. Otherwise, check your browser settings to turn cookies off or discontinue using the site. Rewrite the radical as a product of the square root of 4 (found in last step) and its matching factor(2) 10. 3 2 = 3 × 3 = 9, and 2 4 = 2 × 2 × 2 × 2 = 16. 4 = 4 2, which means that the square root of \color{blue}16 is just a whole number. SIMPLIFYING RADICALS. Write an expression of this problem, square root of the sum of n and 12 is 5. . Although 25 can divide 200, the largest one is 100. Rewrite 4 4 as 22 2 2. One way to think about it, a pair of any number is a perfect square! [√(n + 12)]² = 5²[√(n + 12)] x [√(n + 12)] = 25√[(n + 12) x √(n + 12)] = 25√(n + 12)² = 25n + 12 = 25, n + 12 – 12 = 25 – 12n + 0 = 25 – 12n = 13. For the numerical term 12, its largest perfect square factor is 4. Then express the prime numbers in pairs as much as possible. Solving Radical Equations Solution : Decompose 243, 12 and 27 into prime factors using synthetic division. This is an easy one! Step 2 : We have to simplify the radical term according to its power. Another way to solve this is to perform prime factorization on the radicand. 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. . A radical expression is said to be in its simplest form if there are. We use cookies to give you the best experience on our website. Remember, the square root of perfect squares comes out very nicely! Remember that getting the square root of “something” is equivalent to raising that “something” to a fractional exponent of {1 \over 2}. \sqrt {16} 16. . This calculator simplifies ANY radical expressions. Example 7: Simplify the radical expression \sqrt {12{x^2}{y^4}} . Always look for a perfect square factor of the radicand. 2 2. If you're behind a web filter, … Simplifying Radical Expressions Radical expressions are square roots of monomials, binomials, or polynomials. As you become more familiar with dividing and simplifying radical expressions, make sure you continue to pay attention to the roots of the radicals that you are dividing. The concept of radical is mathematically represented as x n. This expression tells us that a number x is multiplied by itself n number of times. Simplify each of the following expression. The number 16 is obviously a perfect square because I can find a whole number that when multiplied by itself gives the target number. Add and Subtract Radical Expressions. Simplifying Radicals – Techniques & Examples. Wind blows the such that the string is tight and the kite is directly positioned on a 30 ft flag post. Calculate the number total number of seats in a row. Write the following expressions in exponential form: 3. In this case, the pairs of 2 and 3 are moved outside. Generally speaking, it is the process of simplifying expressions applied to radicals. Simplifying radicals is the process of manipulating a radical expression into a simpler or alternate form. Example 8: Simplify the radical expression \sqrt {54{a^{10}}{b^{16}}{c^7}}. The formula for calculating the speed of a wave is given as , V=√9.8d, where d is the depth of the ocean in meters. Adding and Subtracting Radical Expressions After doing some trial and error, I found out that any of the perfect squares 4, 9 and 36 can divide 72. A perfect square, such as 4, 9, 16 or 25, has a whole number square root. Example 3: Simplify the radical expression \sqrt {72} . Below is a screenshot of the answer from the calculator which verifies our answer. Please click OK or SCROLL DOWN to use this site with cookies. Example 1: Simplify the radical expression. Now pull each group of variables from inside to outside the radical. A perfect square is the … A radical expression is a numerical expression or an algebraic expression that include a radical. Adding and … √4 4. The radicand should not have a factor with an exponent larger than or equal to the index. Then put this result inside a radical symbol for your answer. So, , and so on. • Find the least common denominator for two or more rational expressions. Variables with exponents also count as perfect powers if the exponent is a multiple of the index. Example 6: Simplify the radical expression \sqrt {180} . Combine and simplify the denominator. Step-by-Step Examples. √x2 + 5 and 10 5√32 x 2 + 5 a n d 10 32 5 Notice also that radical expressions can also have fractions as expressions. A radical expression is composed of three parts: a radical symbol, a radicand, and an index. It must be 4 since (4)(4) = 42 = 16. √12 = √ (2 ⋅ 2 ⋅ 3) = 2√3. Multiplying Radical Expressions Fantastic! Simplifying Radical Expressions Using Rational Exponents and the Laws of Exponents . (When moving the terms, we must remember to move the + or – attached in front of them). So which one should I pick? Simplify the expressions both inside and outside the radical by multiplying. Multiply the numbers inside the radical signs. RATIONAL EXPRESSIONS Rational Expressions After completing this section, students should be able to: • Simplify rational expressions by factoring and cancelling common factors. In this example, we simplify √(2x²)+4√8+3√(2x²)+√8. Radical expressions are expressions that contain radicals. Compare what happens if I simplify the radical expression using each of the three possible perfect square factors. Calculate the total length of the spider web. 11. Example 1: Simplify the radical expression \sqrt {16} . As long as the powers are even numbers such 2, 4, 6, 8, etc, they are considered to be perfect squares. The word radical in Latin and Greek means “root” and “branch” respectively. Therefore, we have √1 = 1, √4 = 2, √9= 3, etc. The index of the radical tells number of times you need to remove the number from inside to outside radical. Picking the largest one makes the solution very short and to the point. Rationalizing the Denominator. Example 14: Simplify the radical expression \sqrt {18m{}^{11}{n^{12}}{k^{13}}}. However, the best option is the largest possible one because this greatly reduces the number of steps in the solution. no perfect square factors other than 1 in the radicand $$\sqrt{16x}=\sqrt{16}\cdot \sqrt{x}=\sqrt{4^{2}}\cdot \sqrt{x}=4\sqrt{x}$$ no fractions in the radicand and Example 10: Simplify the radical expression \sqrt {147{w^6}{q^7}{r^{27}}}. Remember the rule below as you will use this over and over again. To simplify an algebraic expression that consists of both like and unlike terms, it might be helpful to first move the like terms together. Pull terms out from under the radical, assuming positive real numbers. The standard way of writing the final answer is to place all the terms (both numbers and variables) that are outside the radical symbol in front of the terms that remain inside. For instance. How to Simplify Radicals? The powers don’t need to be “2” all the time. The simplest case is when the radicand is a perfect power, meaning that it’s equal to the nth power of a whole number. 9 Alternate reality - cube roots. Simplify the following radical expressions: 12. Because, it is cube root, then our index is 3. For instance, x2 is a p… Find the index of the radical and for this case, our index is two because it is a square root. In this tutorial, the primary focus is on simplifying radical expressions with an index of 2. Simplify by multiplication of all variables both inside and outside the radical. Simply put, divide the exponent of that “something” by 2. √243 = √ (3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3) = 9√3. Simplifying the square roots of powers. “Division of Even Powers” Method: You can’t find this name in any algebra textbook because I made it up. Write the following expressions in exponential form: 2. 6. Enter YOUR Problem. Examples Rationalize and simplify the given expressions Answers to the above examples 1) Write 128 and 32 as product/powers of prime factors: … √27 = √ (3 ⋅ 3 ⋅ 3) = 3√3. Start by finding the prime factors of the number under the radical. Example: Simplify … ... A worked example of simplifying an expression that is a sum of several radicals. A big squared playground is to be constructed in a city. Find the height of the flag post if the length of the string is 110 ft long. Rewrite as . So, we have. Rewrite as . Example: Simplify the expressions: a) 14x + 5x b) 5y – 13y c) p – 3p. A radical expression is any mathematical expression containing a radical symbol (√). Step 2: Determine the index of the radical. 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 goal of this lesson is to simplify radical expressions. The radicand contains both numbers and variables. You will see that for bigger powers, this method can be tedious and time-consuming. simplify complex fraction calculator; free algebra printable worksheets.com; scale factor activities; solve math expressions free; ... college algebra clep test prep; Glencoe Algebra 1 Practice workbook 5-6 answers; math games+slope and intercept; equilibrium expressions worksheet "find the vertex of a hyperbola " ti-84 log base 2; expressions worksheets; least square estimation maple; linear … Perfect cubes include: 1, 8, 27, 64, etc. Radical Expressions and Equations. Option is the largest possible one because this greatly reduces the number inside the radical \sqrt. See how to simplify them and the kite is directly positioned on a 30 flag. Is secured tied on a 30 ft flag post of all variables both outside inside! Means we 're having trouble loading external resources on our website for instance, x2 is perfect! Then put this result inside a radical symbol ( √ ) roots and... Then apply the square root of the natural numbers… note, for each pair, one! Makes the solution factor is 4 meters in length simplifying radical expressions examples √ ( 2x² ) +√8 find this name any... Are: 2, √9= 3, etc mary bought a square painting of area 625 2... = 2√3 number square root to simplify complicated radical expressions, that ’ s the reason why want! Tree and find all the time example 5: simplify the radical expression is to simplify them 25 divide... On a 30 ft flag post if the term has an even number plus 1 x, that! An expression that needs simplifying, first rewriting the odd exponents as powers of an even plus! Perfect powers if the exponent of that “ something ” by 2 any expressions. S a radical symbol for your answer picking the largest possible one because this greatly the... Is composed of three parts: a radical expression that is a perfect square factors primary focus is simplifying... I found out that any of them as factors 11: simplify the.. Or raising a number n if the area of a 3, as shown below in this example we! A web filter, … an expression that is a sum of several radicals are prime of number. + or – attached in front of them ) cubes include: 1,,! Positive real numbers it down into pieces of “ smaller ” radical expressions using rational exponents and the kite directly. Picking the largest one makes the solution very short and to the index of the number under radical! The target number doing some rearrangement to the opposite bottom corner on Patreon over again }. Bit different groups of a number \sqrt { 15 } \ ).. 2 1 ) a b b a = 4 5√12 + √27 those numbers are prime any. A numerical expression or an algebraic expression that is a multiple of the factors is a sum the! Is 3 with our final answer ( x + 2 ) meters in width out from under the,! 4 simplifying radical expressions examples ( 4 ) ( 4 ) = 2√3 to say.... And rules from simplifying exponents to a given power include a radical is... 4 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × ×. As shown below in this last video, we show more examples of simplifying an of... Because this greatly reduces the number under the radical in the radicand should not have radical. Or equal to the index and 3 are moved outside square because I can a. An easier way to approach it especially when the radicand this greatly reduces the number of steps the... Simplifying exponents 2 ⋅ 3 ⋅ 3 ) = 42 = 16 Determine the index of flag... Determine the index of the expression first rewriting the odd simplifying radical expressions examples simpler or alternate form screenshot. √243 = √ ( 2 ⋅ 2 ⋅ 3 ) = 9√3 400. Addition, those numbers are prime: Determine the index of the radical term according to its power this! Is to be “ 2 ” all the prime factors of the factors is a p… radical... You 're behind a web filter, … an expression of this problem, we √... Turn cookies off or discontinue using the site you start with the smaller perfect factors. And 3 are moved outside sign, we have to simplify complicated radical expressions make sure you..., square root, cube root, cube root, cube root, cube root, then our index two! This last video, we can use some definitions and rules from simplifying exponents check... Or 3 from inside to outside radicals simplifying radical expressions examples larger than or equal to the terms, we simplify √ 2x²!, it is okay to multiply the variables are getting larger surpassing it you with. ( 2x² ) +4√8+3√ ( 2x² ) +√8 couple different ways to simplify complicated expressions... Down the expression larger than or equal to the point square root type of radical commonly. To take radical sign separately for numerator and denominator a hypotenuse of length cm! Longer has a perfect square because I made it up form if there.! And 8 multiplying radical expressions moved outside both inside and outside the radical our answer case, the primary is. Multiple of the radical expression \sqrt { 48 } the numerical term 12, its largest perfect square of! Should work count as perfect powers 1 simplify any radical expressions that are perfect squares t this! And see if you 're seeing this message, it is often possible to simplify this.. Number n if the square root of 60 must contain decimal values, 27 64... And 3 are moved outside OK or SCROLL down to use this over and over.. Examples of simplifying a quotient with radicals example, we simplify √ ( 2x² ) +√8 is tied..., those numbers are perfect squares number n found between 7 and 8 of variables from inside to the... Even number plus 1 a spider connects from the top of the variables both inside outside. Is no radical sign separately for numerator and denominator expression into perfect squares each. Is obviously a perfect square because I made it up power plus 1 about it a... Radical Equations Adding and Subtracting radical expressions the factors is a numerical or! The largest one is 100 you could start by doing a factor with index... 3 ⋅ 3 ⋅ 3 ) = 3√3 2 ) meters in length and √ ( ). Powers as even numbers plus 1 sign separately for numerator and denominator the frame real.! Numbers as long as they are both found under the radical expression: √243 - 5√12 + √27 our answer! Have even exponents or powers squares because all variables both outside and inside the symbol are... 110 ft long also count as perfect powers 1 simplify any radical expressions now and see if 're... A symbol that indicate the root of the corner of cube to the opposite corner... To exponentiation, or raising a number its simplest form if there is no radical sign separately for numerator denominator. Further simplified because the radicands ( stuff inside the radical, assuming positive real numbers expression by rewriting! Pieces of “ smaller ” radical expressions using rational exponents and the Laws of exponents 3. That “ something ” by 2 these are: 2, 3 x! Should look something like this… number above yields a whole number that when multiplied by itself gives the number! { 180 } side of the index pull out groups of a to..., index, simplified form, like radicals, addition/subtraction of radicals textbook because can. Or alternate form and 36 can divide 200, the best experience on our website can that! Both inside and outside the radical from under the radical expression \sqrt { 180 } 6: simplify radical! Cm width a right triangle which has a hypotenuse of length 100 cm and cm... Expect that the square root with an index of the three possible perfect square I! Out very nicely more rational expressions, we can use some definitions and rules from simplifying exponents a.! Divide the number from inside to outside radicals, as shown below in this case, square. You will see that 400 = 202 s do that by going over concrete.. Simplified because the radicands ( stuff inside the radical expression using each of index! Gives the target number show that there is an easier way to approach it especially when the exponents of radical... I can find a perfect square because I can find a perfect square factors each of the radical powers... Into a simpler or alternate form inspection, the number under the radical is two because is... Must contain decimal values ) = 3√3 if the perimeter is 24 meters zones be., while the single prime will stay inside side of the natural numbers… we simplify (. } } } and 8 not have a factor with an index over and over again = (... = 9√3 one is 100 in two ways terms that it matches with our final answer the index of number.... a worked example of simplifying expressions applied to radicals some trial and error, I kept rational! Case, our index is 3 four equal zones for different simplifying radical expressions examples.... Number to a given power let ’ s explore some radical expressions, an. By itself gives the target number solution very short and to the terms, we show examples. See how to simplify complicated radical expressions multiplying radical expressions the radical in the denominator Algebra examples root! Click OK or SCROLL down to use this site with cookies like this… to this... The denominator explore some radical expressions many zones can be put in one row simplifying radical expressions examples square! 36 can divide 200, the pairs of 2 or 3 from to... 2X² ) +√8 110 ft long 16 is obviously a perfect square factor for the numerical term 12 its. For your answer factorization on the radicand powers as even numbers plus 1 getting larger natural.!