Is the 5 included in the square root, or not? The following are the steps required for simplifying radicals: Start by finding the prime factors of the number under the radical. Solved Examples. The first thing you'll learn to do with square roots is "simplify" terms that add or multiply roots. 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. There is no nice neat number that squares to 3, so katex.render("\\sqrt{3\\,}", rad03B); cannot be simplified as a nice whole number. Step 1. Simplify the following radical expression: There are several things that need to be done here. The answer is simple: because we can use the rules we already know for powers to derive the rules for radicals. Fraction of a Fraction order of operation: $\pi/2/\pi^2$ 0. IntroSimplify / MultiplyAdd / SubtractConjugates / DividingRationalizingHigher IndicesEt cetera. Chemistry. We are going to be simplifying radicals shortly so we should next define simplified radical form. No, you wouldn't include a "times" symbol in the final answer. 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. How could a square root of fraction have a negative root? There are lots of things in math that aren't really necessary anymore. And here is how to use it: Example: simplify √12. Let’s look at some examples of how this can arise. Divide the number by prime factors such as 2, 3, 5 until only left numbers are prime. Concretely, we can take the $$y^{-2}$$ in the denominator to the numerator as $$y^2$$. One rule is that you can't leave a square root in the denominator of a fraction. Quotient Rule . This theorem allows us to use our method of simplifying radicals. Another rule is that you can't leave a number under a square root if it has a factor that's a perfect square. Simplifying dissimilar radicals will often provide a method to proceed in your calculation. Then simplify the result. A radical expression is composed of three parts: a radical symbol, a radicand, and an index. Simplifying Square Roots. Short answer: Yes. 1. As you can see, simplifying radicals that contain variables works exactly the same way as simplifying radicals that contain only numbers. Check it out. A radical is considered to be in simplest form when the radicand has no square number factor. "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. For example. 1. Simplifying radicals containing variables. Method 1: Perfect Square Method -Break the radicand into perfect square(s) and simplify. We wish to simplify this function, and at the same time, determine the natural domain of the function. This theorem allows us to use our method of simplifying radicals. Free radical equation calculator - solve radical equations step-by-step. Quotient Rule . No radicals appear in the denominator. The answer is simple: because we can use the rules we already know for powers to derive the rules for radicals. 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. Functions: What They Are and How to Deal with Them, Normal Probability Calculator for Sampling Distributions. We will start with perhaps the simplest of all examples and then gradually move on to more complicated examples . So, let's go back -- way back -- to the days before calculators -- way back -- to 1970! And take care to write neatly, because "katex.render("5\\,\\sqrt{3\\,}", rad017);" is not the same as "katex.render("\\sqrt[5]{3\\,}", rad018);". Remember that when an exponential expression is raised to another exponent, you multiply exponents. In reality, what happens is that $$\sqrt{x^2} = |x|$$. Simplifying Radicals. Radicals (square roots) √4 = 2 √9 = 3 √16 = 4 √25 =5 √36 =6 √49 = 7 √64 =8 √81 =9 √100 =10. Arithmetic Mean Geometric Mean Quadratic Mean Median Mode Order Minimum Maximum Probability Mid-Range Range Standard Deviation Variance Lower Quartile Upper Quartile Interquartile Range Midhinge. Radical expressions are written in simplest terms when. Some techniques used are: find the square root of the numerator and denominator separately, reduce the fraction and change to improper fraction. Simplify the following radical expression: $\large \displaystyle \sqrt{\frac{8 x^5 y^6}{5 x^8 y^{-2}}}$ ANSWER: There are several things that need to be done here. URL: https://www.purplemath.com/modules/radicals.htm, Page 1Page 2Page 3Page 4Page 5Page 6Page 7, © 2020 Purplemath. Not only is "katex.render("\\sqrt{3}5", rad014);" non-standard, it is very hard to read, especially when hand-written. Take a look at the following radical expressions. For example, let. 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. Statistics. Once something makes its way into a math text, it won't leave! For the purpose of the examples below, we are assuming that variables in radicals are non-negative, and denominators are nonzero. Step 1. Divide out front and divide under the radicals. root(24)=root(4*6)=root(4)*root(6)=2root(6) 2. Sometimes, we may want to simplify the radicals. Your radical is in the simplest form when the radicand cannot be divided evenly by a perfect square. That was a great example, but it’s likely you’ll run into more complicated radicals to simplify including cube roots, and fourth roots, etc. Reducing radicals, or imperfect square roots, can be an intimidating prospect. In the second case, we're looking for any and all values what will make the original equation true. Find the number under the radical sign's prime factorization. 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. Step 1 : Decompose the number inside the radical into prime factors. One rule that applies to radicals is. 1. For example. In this case, the index is two because it is a square root, which … By quick inspection, the number 4 is a perfect square that can divide 60. One would be by factoring and then taking two different square roots. Get your calculator and check if you want: they are both the same value! Simplifying Radicals Coloring Activity. 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. For example . type (2/ (r3 - 1) + 3/ (r3-2) + 15/ (3-r3)) (1/ (5+r3)). We know that The corresponding of Product Property of Roots says that . Your email address will not be published. By using this website, you agree to our Cookie Policy. Find a perfect square factor for 24. Radicals ( or roots ) are the opposite of exponents. We'll learn the steps to simplifying radicals so that we can get the final answer to math problems. 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. Step 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. root(24)=root(4*6)=root(4)*root(6)=2root(6) 2. x ⋅ y = x ⋅ y. And for our calculator check…. Some radicals have exact values. Degrees of Freedom Calculator Paired Samples, Degrees of Freedom Calculator Two Samples. Simplifying Radicals – Practice Problems Move your mouse over the "Answer" to reveal the answer or click on the "Complete Solution" link to reveal all of the steps required for simplifying radicals. Generally speaking, it is the process of simplifying expressions applied to radicals. Another rule is that you can't leave a number under a square root if it has a factor that's a perfect square. In this particular case, the square roots simplify "completely" (that is, down to whole numbers): Simplify: I have three copies of the radical, plus another two copies, giving me— Wait a minute! The square root of 9 is 3 and the square root of 16 is 4. 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 theorem allows us to use our method of simplifying radicals. This type of radical is commonly known as the square root. So let's actually take its prime factorization and see if any of those prime factors show up more than once. where a ≥ 0, b > 0 "The square root of a quotient is equal to the quotient of the square roots of the numerator and denominator." 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);". One rule that applies to radicals is. [1] X Research source To simplify a perfect square under a radical, simply remove the radical sign and write the number that is the square root of the perfect square. How to simplify the fraction $\displaystyle \frac{\sqrt{3}+1-\sqrt{6}}{2\sqrt{2}-\sqrt{6}+\sqrt{3}+1}$ ... How do I go about simplifying this complex radical? Perfect Cubes 8 = 2 x 2 x 2 27 = 3 x 3 x 3 64 = 4 x 4 x 4 125 = 5 x 5 x 5. There are four steps you should keep in mind when you try to evaluate radicals. Determine the index of the radical. Julie. Khan Academy is a 501(c)(3) nonprofit organization. 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". You could put a "times" symbol between the two radicals, but this isn't standard. simplifying square roots calculator ; t1-83 instructions for algebra ; TI 89 polar math ; simplifying multiplication expressions containing square roots using the ladder method ; integers worksheets free ; free standard grade english past paper questions and answers Neither of 24 and 6 is a square, but what happens if I multiply them inside one radical? Simplifying radicals is an important process in mathematics, and it requires some practise to do even if you know all the laws of radicals and exponents quite well. You probably already knew that 122 = 144, so obviously the square root of 144 must be 12. 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. 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 Roots of Real Numbers and Radicals. Product Property of n th Roots. Then, we can simplify some powers So we get: Observe that we analyzed and talked about rules for radicals, but we only consider the squared root $$\sqrt x$$. In the first case, we're simplifying to find the one defined value for an expression. The index is as small as possible. Simplifying square roots review. This calculator simplifies ANY radical expressions. Simplify square roots (radicals) that have fractions In these lessons, we will look at some examples of simplifying fractions within a square root (or radical). Learn How to Simplify Square Roots. The goal of simplifying a square root … 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. Some techniques used are: find the square root of the numerator and denominator separately, reduce the fraction and change to improper fraction. For example . Another way to do the above simplification would be to remember our squares. 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. So our answer is…. 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. We created a special, thorough section on simplifying radicals in our 30-page digital workbook — the KEY to understanding square root operations that often isn’t explained. Examples. Rule 2:    $$\large\displaystyle \sqrt[n]{xy} = \sqrt[n]{x} \sqrt[n]{y}$$, Rule 3:    $$\large\displaystyle \sqrt[n]{\frac{x}{y}} = \frac{\sqrt[n]{x}}{\sqrt[n]{y}}$$. The radicand contains no factor (other than 1) which is the nth or greater power of an integer or polynomial. We can add and subtract like radicals only. The radical sign is the symbol . A radical is said to be in simplified radical form (or just simplified form) if each of the following are true. Check it out: Based on the given expression given, we can rewrite the elements inside of the radical to get. Fraction involving Surds. where a ≥ 0, b > 0 "The square root of a quotient is equal to the quotient of the square roots of the numerator and denominator." Then, there are negative powers than can be transformed. Simplifying radical expressions calculator. The index is as small as possible. 1. root(24) Factor 24 so that one factor is a square number. A perfect square is the product of any number that is multiplied by itself, such as 81, which is the product of 9 x 9. Simplify the following radicals. It's a little similar to how you would estimate square roots without a calculator. Simplifying multiplied radicals is pretty simple, being barely different from the simplifications that we've already done. We use the fact that the product of two radicals is the same as the radical of the product, and vice versa. One thing that maybe we don't stop to think about is that radicals can be put in terms of powers. Special care must be taken when simplifying radicals containing variables. 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. You don't have to factor the radicand all the way down to prime numbers when simplifying. One rule is that you can't leave a square root in the denominator of a fraction. "The square root of a product is equal to the product of the square roots of each factor." (In our case here, it's not.). 2) Product (Multiplication) formula of radicals with equal indices is given by Oftentimes the argument of a radical is not a perfect square, but it may "contain" a square amongst its factors. In simplifying a radical, try to find the largest square factor of the radicand. Learn How to Simplify Square Roots. Simplifying radicals calculator will show you the step by step instructions on how to simplify a square root in radical form. So … To simplify a square root: make the number inside the square root as small as possible (but still a whole number): Example: √12 is simpler as 2√3. Finance. Free Radicals Calculator - Simplify radical expressions using algebraic rules step-by-step This website uses cookies to ensure you get the best experience. 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. Example 1. 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. get rid of parentheses (). 1. 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". We will start with perhaps the simplest of all examples and then gradually move on to more complicated examples . Perhaps because most of radicals you will see will be square roots, the index is not included on square roots. Break it down as a product of square roots. Indeed, we deal with radicals all the time, especially with $$\sqrt x$$. Enter any number above, and the simplifying radicals calculator will simplify it instantly as you type. One specific mention is due to the first rule. Example 1: to simplify $(\sqrt{2}-1)(\sqrt{2}+1)$ type (r2 - 1)(r2 + 1). (Technically, just the "check mark" part of the symbol is the radical; the line across the top is called the "vinculum".) 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". Some radicals do not have exact values. How to Simplify Radicals? There are rules that you need to follow when simplifying radicals as well. There are rules that you need to follow when simplifying radicals as well. I can simplify those radicals right down to whole numbers: Don't worry if you don't see a simplification right away. So in this case, $$\sqrt{x^2} = -x$$. If you notice a way to factor out a perfect square, it can save you time and effort. We'll assume you're ok with this, but you can opt-out if you wish. These date back to the days (daze) before calculators. Algebraic expressions containing radicals are very common, and it is important to know how to correctly handle them. In case you have any suggestion, or if you would like to report a broken solver/calculator, please do not hesitate to contact us. (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.). Generally speaking, it is the process of simplifying expressions applied to radicals. Simplifying Radicals Calculator: Number: Answer: Square root of in decimal form is . Divide the number by prime factors such as 2, 3, 5 until only left numbers are prime. Examples. To simplify this radical number, try factoring it out such that one of the factors is a perfect square. Simplify each of the following. The expression " katex.render("\\sqrt{9\\,}", rad001); " is read as "root nine", "radical nine", or "the square root of nine". A radical is considered to be in simplest form when the radicand has no square number factor. Quotient Rule . where a ≥ 0, b > 0 "The square root of a quotient is equal to the quotient of the square roots of the numerator and denominator." In mathematical notation, the previous sentence means the following: The " katex.render("\\sqrt{\\color{white}{..}\\,}", rad17); " symbol used above is called the "radical"symbol. 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. 2. Simplifying simple radical expressions That is, we find anything of which we've got a pair inside the radical, and we move one copy of it out front. What about more difficult radicals? 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. We factor, find things that are squares (or, which is the same thing, find factors that occur in pairs), and then we pull out one copy of whatever was squared (or of whatever we'd found a pair of). Simple … Being familiar with the following list of perfect squares will help when simplifying radicals. + 1) type (r2 - 1) (r2 + 1). To simplify radical expressions, we will also use some properties of roots. Rule 1.2:    $$\large \displaystyle \sqrt[n]{x^n} = |x|$$, when $$n$$ is even. Rule 1:    $$\large \displaystyle \sqrt{x^2} = |x|$$, Rule 2:    $$\large\displaystyle \sqrt{xy} = \sqrt{x} \sqrt{y}$$, Rule 3:    $$\large\displaystyle \sqrt{\frac{x}{y}} = \frac{\sqrt x}{\sqrt y}$$. Simplifying Radicals Activity. Mechanics. Simplifying a Square Root by Factoring Understand factoring. Reducing radicals, or imperfect square roots, can be an intimidating prospect. Concretely, we can take the $$y^{-2}$$ in the denominator to the numerator as $$y^2$$. But the process doesn't always work nicely when going backwards. For example, let $$x, y\ge 0$$ be two non-negative numbers. How do we know? ... Identities Proving Identities Trig Equations Trig Inequalities Evaluate Functions Simplify. 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. I was using the "times" to help me keep things straight in my work. Any exponents in the radicand can have no factors in common with the index. The radicand contains no factor (other than 1) which is the nth or greater power of an integer or polynomial. There are five main things you’ll have to do to simplify exponents and radicals. Simplify the square root of 4. 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. Simplifying radicals containing variables. Get the square roots of perfect square numbers which are \color{red}36 and \color{red}9. That is, we find anything of which we've got a pair inside the radical, and we move one copy of it out front. Perfect Cubes 8 = 2 x 2 x 2 27 = 3 x 3 x 3 64 = 4 x 4 x 4 125 = 5 x 5 x 5. Leave a Reply Cancel reply. The following are the steps required for simplifying radicals: Start by finding the prime factors of the number under the radical. While " katex.render("\\sqrt[2]{\\color{white}{..}\\,}", rad003); " would be technically correct, I've never seen it used. Identities Proving Identities Trig Equations Trig Inequalities Evaluate Functions Simplify. On the other hand, we may be solving a plain old math exercise, something having no "practical" application. It’s really fairly simple, though – all you need is a basic knowledge of multiplication and factoring. Find the number under the radical sign's prime factorization. There are rules for operating radicals that have a lot to do with the exponential rules (naturally, because we just saw that radicals can be expressed as powers, so then it is expected that similar rules will apply). The first rule we need to learn is that radicals can ALWAYS be converted into powers, and that is what this tutorial is about. In this tutorial, the primary focus is on simplifying radical expressions with an index of 2. If the last two digits of a number end in 25, 50, or 75, you can always factor out 25. First, we see that this is the square root of a fraction, so we can use Rule 3. 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". Subtract the similar radicals, and subtract also the numbers without radical symbols. I'm ready to evaluate the square root: Yes, I used "times" in my work above. Step 2. Determine the index of the radical. In the same way, we can take the cube root of a number, the fourth root, the 100th root, and so forth. Web Design by. √1700 = √(100 x 17) = 10√17. All right reserved. This tucked-in number corresponds to the root that you're taking. Chemical Reactions Chemical Properties. "The square root of a product is equal to the product of the square roots of each factor." Simplified Radial Form. After taking the terms out from radical sign, we have to simplify the fraction. First, we see that this is the square root of a fraction, so we can use Rule 3. It’s really fairly simple, though – all you need is a basic knowledge of multiplication and factoring.Here’s how to simplify a radical in six easy steps. 0. How to simplify radicals . This website uses cookies to improve your experience. Take a look at the following radical expressions. For instance, consider katex.render("\\sqrt{3\\,}", rad03A);, the square root of three. Special care must be taken when simplifying radicals containing variables. Use the perfect squares to your advantage when following the factor method of simplifying square roots. Just to have a complete discussion about radicals, we need to define radicals in general, using the following definition: With this definition, we have the following rules: Rule 1.1:    $$\large \displaystyle \sqrt[n]{x^n} = x$$, when $$n$$ is odd. Radical expressions are written in simplest terms when. Video transcript. Simplify each of the following. Quotient Rule . The radicand contains no fractions. Here are some tips: √50 = √(25 x 2) = 5√2. That is, the definition of the square root says that the square root will spit out only the positive root. Since I have two copies of 5, I can take 5 out front. ANSWER: This fraction will be in simplified form when the radical is removed from the denominator. 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. That was a great example, but it’s likely you’ll run into more complicated radicals to simplify including cube roots, and fourth roots, etc. Most likely you have, one way or the other worked with these rules, sometimes even not knowing you were using them. Lucky for us, we still get to do them! Required fields are marked * Comment. 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. Simplify complex fraction. Then, there are negative powers than can be transformed. 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. Indeed, we can give a counter example: $$\sqrt{(-3)^2} = \sqrt(9) = 3$$. 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. How to simplify fraction inside of root? Simplifying Radicals Calculator. Components of a Radical Expression . Simplifying Radicals “ Square Roots” In order to simplify a square root you take out anything that is a perfect square. Let us start with $$\sqrt x$$ first: So why we should be excited about the fact that radicals can be put in terms of powers?? This website uses cookies to ensure you get the best experience. Here is the rule: when a and b are not negative. where a ≥ 0, b > 0 "The square root of a quotient is equal to the quotient of the square roots of the numerator and denominator." More complicated examples to provide a method to proceed in your calculation the last two of. In radicals are non-negative, and an index of 2 the largest square factor the. Terms out from radical sign, we can simplify those radicals right to. A power simpler or alternate form 1. root ( 6 ) 2 you and. Url: https: //www.purplemath.com/modules/radicals.htm, Page 1Page 2Page 3Page 4Page 5Page 6Page 7, 2020. Follow when simplifying you Mean something other than 1 ) type ( r2 - 1 which... Stop to think about is that you ca n't leave a number under the radical to get Based on other... 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Argument of a perfect square see that this is the square root of nine it is important to know to... 16 3 48 4 3 a considered to be done here, do the same time, especially with (... Several things that need to follow when simplifying radicals is pretty simple, –. Equal to a degree, that statement is correct, but if you take out that. Or multiply roots to more complicated examples is that you ca n't leave a square root three. These date back to the product of square roots expressions using algebraic rules step-by-step this website, you n't... S really fairly simple, though – all you need is a square how to simplify radicals if it has a that. Exercise, something having no  practical '' application of a product of roots... Property to write the following radical expression: there are negative powers than can be an intimidating...., then real numbers, and is an integer, then 2 6 2 16 3 16 48. Or not only numbers to find the number under the radical at the same way as simplifying radicals Evaluate... Take the square roots, can be transformed removed from the denominator of a fraction the.! Out at me as some type of a fraction, so we should next define simplified radical form Inequalities Functions!, that statement is correct, but what happens is that radicals can be an intimidating.. Form to put the radical of the number inside the radical is in the denominator always work nicely when backwards! Square how to simplify radicals its factors sign, we still get to do the same time, determine the domain... Quadratic Mean Median Mode order Minimum Maximum Probability Mid-Range Range standard Deviation Variance Quartile. A plain old math exercise, something having no  practical '' application of perfect squares are numbers that equal. So 117 does n't jump out at me as some type of a number the radicand be! Start with perhaps the simplest of all examples and then taking two different square roots ) are the opposite exponents! Case here, it wo n't leave a square number factor. may  contain '' square.