The Product Raised to a Power Rule and the Quotient Raised to a Power Rule can be used to simplify radical expressions as long as the roots of the radicals are the same. When you're multiplying radicals together, you can combine the two into one radical expression. Answer D contains a problem and answer pair that is incorrect. Dividing Radical Expressions. The Product Raised to a Power Rule and the Quotient Raised to a Power Rule can be used to simplify radical expressions as long as the roots of the radicals are the same. The expression  is the same as , but it can also be simplified further. Module 4: Dividing Radical Expressions Recall the property of exponents that states that m m m a a b b ⎛⎞ =⎜⎟ ⎝⎠. For example, while you can think of as equivalent to since both the numerator and the denominator are square roots, notice that you cannot express as . Are you sure you want to remove #bookConfirmation# Multiplying, dividing, adding, subtracting negative numbers all in one, tic tac toe factoring method, algebra worksheet puzzles, solving second order differential equations by simulation in matlab of motor bhavior equation, least common multiple with variables, rules when adding & subtracting integers, solving linear equations two variables … Be looking for powers of 4 in each radicand. Quotient Raised to a Power Rule. Each variable is considered separately. Note that the roots are the same—you can combine square roots with square roots, or cube roots with cube roots, for example. Here we cover techniques using the conjugate. Simplify each radical, if possible, before multiplying. Dividing radicals with variables is the same as dividing them without variables . A common way of dividing the radical expression is to have the denominator that contain no radicals. You multiply radical expressions that contain variables in the same manner. The two radicals have different roots, so you cannot multiply the product of the radicands and put it under the same radical sign. How would the expression change if you simplified each radical first, before multiplying? It includes simplifying radicals with roots greater than 2. As with multiplication, the main idea here is that sometimes it makes sense to divide and then simplify, and other times it makes sense to simplify and then divide. All rights reserved. A) Correct. This problem does not contain any errors. Look for perfect cubes in the radicand. Correct. Answer D contains a problem and answer pair that is incorrect. If a and b are unlike terms, then the conjugate of a + b is a – b, and the conjugate of a – b is a + b. We can drop the absolute value signs in our final answer because at the start of the problem we were told , . dividing radical expressions worksheets, multiplying and dividing … Divide and simplify radical expressions that contain a single term. In both cases, you arrive at the same product, Look for perfect cubes in the radicand. 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. There are five main things you’ll have to do to simplify exponents and radicals. B) Problem:  Answer: Incorrect. The end result is the same, . (Express your answer in simplest radical form) Use the rule  to multiply the radicands. This rule states that the product of two or more numbers raised to a power is equal to the product of each number raised to the same power. What if you found the quotient of this expression by dividing within the radical first, and then took the cube root of the quotient? Using what you know about quotients, you can rewrite the expression as, Incorrect. In both problems, the Product Raised to a Power Rule is used right away and then the expression is simplified. Which one of the following problem and answer pairs is incorrect? By the way, concerning Multiplying and Dividing Radicals Worksheets, we have collected several related photos to complete your references. You correctly took the square roots of  and , but you can simplify this expression further. This should be a familiar idea. Using the Product Raised to a Power Rule, you can take a seemingly complicated expression. get rid of parentheses (). In this case, notice how the radicals are simplified before multiplication takes place. The two radicals that are being multiplied have the same root (3), so they can be multiplied together underneath the same radical sign. This is an advanced look at radicals. You simplified , not . That's a mathematical symbols way of saying that when the index is even there can be no negative number in the radicand, but when the index is odd, there can be. What can be multiplied with so the result will not involve a radical? Quiz Multiplying Radical Expressions, Next Recall that the Product Raised to a Power Rule states that, As you did with multiplication, you will start with some examples featuring integers before moving on to more complex expressions like, That was a lot of effort, but you were able to simplify using the. This problem does not contain any errors; You can use the same ideas to help you figure out how to simplify and divide radical expressions. Incorrect. Example Questions. Multiplying And Dividing Radicals Worksheets admin April 22, 2020 Some of the worksheets below are Multiplying And Dividing Radicals Worksheets, properties of radicals, rules for simplifying radicals, radical operations practice exercises, rationalize the denominator and multiply with radicals worksheet with … As long as the roots of the radical expressions are the same, you can use the Product Raised to a Power Rule to multiply and simplify. For example, while you can think of  as equivalent to  since both the numerator and the denominator are square roots, notice that you cannot express  as . C) Incorrect. Let’s take another look at that problem. You correctly took the square roots of  and , but you can simplify this expression further. When dividing radical expressions, we use the quotient rule to help solve them. Factor the number into its prime factors and expand the variable(s). The correct answer is . The expression  is the same as , but it can also be simplified further. That was a lot of effort, but you were able to simplify using the Quotient Raised to a Power Rule. The radicand contains no factor (other than 1) which is the nth or greater power of an integer or polynomial. Quiz: Dividing Rational Expressions Adding and Subtracting Rational Expressions Examples of Rational Expressions What is the sum of the polynomials 3a2b + 2a2b2 plus -ab, dividing variables worksheet, common denominator calculator, first in math cheats, mathpoem, foil solver math, Printable Formula Chart. from your Reading List will also remove any C) Problem:  Answer: Incorrect. You simplified , not . Look for perfect squares in the radicand, and rewrite the radicand as the product of two factors. Multiplying and dividing radicals makes use of the "Product Rule" and the "Quotient Rule" as seen at the right. That was a more straightforward approach, wasn’t it? Multiplying and dividing radical expressions worksheet with answers Collection. ... (Assume all variables are positive.) Incorrect. Now let’s turn to some radical expressions containing variables. The "n" simply means that the index could be any value.Our examples will be using the index to be 2 (square root). If you think of the radicand as a product of two factors (here, thinking about 64 as the product of 16 and 4), you can take the square root of each factor and then multiply the roots. ... Equations for calculating, algebra 2 practice tests, radicals with variables. The same is true of roots: . Today we deliver you various awesome photos that we collected in case you need more example, for today we are focused related with Multiplying and Dividing Radicals Worksheets. Look for perfect square factors in the radicand, and rewrite the radicand as a product of factors. The correct answer is . Since all the radicals are fourth roots, you can use the rule  to multiply the radicands. According to the Product Raised to a Power Rule, this can also be written , which is the same as , since fractional exponents can be rewritten as roots. A) Problem:  Answer: 20 Incorrect. Using the Product Raised to a Power Rule, you can take a seemingly complicated expression, , and turn it into something more manageable,. Using what you know about quotients, you can rewrite the expression as , simplify it to , and then pull out perfect squares. The terms in this expression are both cube roots, but I can combine them only if they're the cube roots of the same value. and any corresponding bookmarks? If n is even, and a ≥ 0, b > 0, then. We just have to work with variables as well as numbers. Identify perfect cubes and pull them out of the radical. What if you found the quotient of this expression by dividing within the radical first, and then took the cube root of the quotient? Let’s start with a quantity that you have seen before, This should be a familiar idea. That's a mathematical symbols way of saying that when the index is even there can be no negative number in the radicand, but when the index is odd, there can be. To rationalize this denominator, the appropriate fraction with the value 1 is , since that will eliminate the radical in the denominator, when used as follows: Note we elected to find 's principal root. Incorrect. Definition: If \(a\sqrt b + c\sqrt d \) is a radical expression, then the conjugate is \(a\sqrt b - c\sqrt d \). Quiz & Worksheet - Dividing Radical Expressions | Study.com #117518 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). ... , divide, dividing radicals, division, index, Multiplying and Dividing Radicals, multiplying radicals, radical, rationalize, root. This process is called rationalizing the denominator. The two radicals that are being multiplied have the same root (3), so they can be multiplied together underneath the same radical sign. Look for perfect cubes in the radicand, and rewrite the radicand as a product of factors. Free printable worksheets with answer keys on Radicals, Square Roots (ie no variables)includes visual aides, model problems, exploratory activities, practice problems, and an online component Answer D contains a problem and answer pair that is incorrect. The Product Rule states that the product of two or more numbers raised to a power is equal to the product of each number raised to the same power. But you can’t multiply a square root and a cube root using this rule. The conjugate of is . This worksheet has model problems worked out, step by step as well as 25 scaffolded questions that start out relatively easy and end with some real challenges. D) Problem:  Answer: Correct. Right now, they aren't. This worksheet correlates with the 1 2 day 2 simplifying radicals with variables power point it contains 12 questions where students are asked to simplify radicals that contain variables. Newer Post Older Post Home. So, for the same reason that , you find that . You can do more than just simplify radical expressions. You can multiply and divide them, too. This property can be used to combine two radicals … Adding and subtracting radicals is much like combining like terms with variables. Notice that the process for dividing these is the same as it is for dividing integers. The simplified form is . Students will practice dividing square roots (ie radicals). Multiply and simplify radical expressions that contain a single term. So I'll simplify the radicals first, and then see if I can go any further. One helpful tip is to think of radicals as variables, and treat them the same way. Use the Quotient Raised to a Power Rule to rewrite this expression. For example, while you can think of, Correct. Use the rule  to create two radicals; one in the numerator and one in the denominator. Then, using the greatest common factor, … Quiz Dividing Radical Expressions. Division with radicals is very similar to multiplication, if we think about division as reducing fractions, we can reduce the coefficients outside the radicals and reduce the values inside the radicals to get our final solution. Removing #book# simplifying radicals with variables examples, LO: I can simplify radical expressions including adding, subtracting, multiplying, dividing and rationalizing denominators. CliffsNotes study guides are written by real teachers and professors, so no matter what you're studying, CliffsNotes can ease your homework headaches and help you score high on exams. When dividing radical expressions, use the quotient rule. Multiplying and dividing radicals. Now when dealing with more complicated expressions involving radicals, we employ what is known as the conjugate. Let’s start with a quantity that you have seen before,. Dividing radical is based on rationalizing the denominator.Rationalizing is the process of starting with a fraction containing a radical in its denominator and determining fraction with no radical in … Well, what if you are dealing with a quotient instead of a product? Adding Subtracting Multiplying Radicals Worksheets Dividing Radicals Worksheets Algebra 1 Algebra 2 Square Roots Radical Expressions Introduction Topics: Simplifying radical expressions Simplifying radical expressions with variables Adding radical expressions Multiplying radical expressions Removing radicals from the … I usually let my students play in pairs or groups to review for a test. Recall that the Product Raised to a Power Rule states that . Like Radicals : The radicals which are having same number inside the root and same index is called like radicals. You have applied this rule when expanding expressions such as (ab)x to ax • bx; now you are going to amend it to include radicals as well. Simplify each radical. This problem does not contain any errors; . When dividing radical expressions, the rules governing quotients are similar: . Whichever order you choose, though, you should arrive at the same final expression. There's a similar rule for dividing two radical expressions. The Product Rule states that the product of two or more numbers raised to a power is equal to the product of each number raised to the same power. Answer D contains a problem and answer pair that is incorrect. Since both radicals are cube roots, you can use the rule  to create a single rational expression underneath the radical. If you have one square root divided by another square root, you can combine them together with division inside one square root. Simplify each expression by factoring to find perfect squares and then taking … Since both radicals are cube roots, you can use the rule, 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. In this second case, the numerator is a square root and the denominator is a fourth root. The correct answer is . For any numbers a and b and any integer x: For any numbers a and b and any positive integer x: The Product Raised to a Power Rule is important because you can use it to multiply radical expressions. The students help each other work the problems. Since  is not a perfect cube, it has to be rewritten as . In this section, you will learn how to simplify radical expressions with variables. You correctly took the square roots of. A Variable is a symbol for a number we don't know yet. Variables with Exponents How to Multiply and Divide them What is a Variable with an Exponent? This problem does not contain any errors; . (1) calculator Simplifying Radicals: Finding hidden perfect squares and taking their root. When dividing radical expressions, use the quotient rule. An expression with a radical in its denominator should be simplified into one without a radical in its denominator. For the purpose of the examples below, we are assuming that variables in radicals are non-negative, and denominators are nonzero. cals are simplified and all like radicals or like terms have been combined. You can simplify this expression even further by looking for common factors in the numerator and denominator. Radicals Simplifying Radicals … Directions: Divide the radicals below. Since, Identify and pull out powers of 4, using the fact that, Since all the radicals are fourth roots, you can use the rule, Now that the radicands have been multiplied, look again for powers of 4, and pull them out. The correct answer is . For any real numbers a and b (b ≠ 0) and any positive integer x: As you did with multiplication, you will start with some examples featuring integers before moving on to more complex expressions like . Correct. In this example, we simplify √(2x²)+4√8+3√(2x²)+√8. Incorrect. The Quotient Raised to a Power Rule states that . When radicals (square roots) include variables, they are still simplified the same way. Identify and pull out powers of 4, using the fact that . Incorrect. We can add and subtract like radicals … D) Incorrect. The number coefficients are reduced the same as in simple fractions. Free Algebra … This algebra video tutorial explains how to multiply radical expressions with variables and exponents. ©o 6KCuAtCav QSMoMfAtIw0akrLeD nLrLDCj.r m 0A0lsls 1r6i4gwh9tWsx 2rieAsKeLrFvpe9dc.c G 3Mfa0dZe7 UwBixtxhr AIunyfVi2nLimtqel bAmlCgQeNbarwaj w1Q.V-6-Worksheet by Kuta Software LLC Answers to Multiplying and Dividing Radicals Look for perfect squares in the radicand. This is accomplished by multiplying the expression by a fraction having the value 1, in an appropriate form. Making sense of a string of radicals may be difficult. The quotient rule states that a radical involving a quotient is equal to the quotients of two radicals. We can drop the absolute value signs in our final answer because at the start of the problem we were told. An exponent (such as the 2 in x 2) says how many times to use the variable in a multiplication. The same is true of roots. The correct answer is . So, this problem and answer pair is incorrect. You have applied this rule when expanding expressions such as (. Incorrect. It is usually a letter like x or y. (Remember that the order you choose to use is up to you—you will find that sometimes it is easier to multiply before simplifying, and other times it is easier to simplify before multiplying. A worked example of simplifying an expression that is a sum of several radicals. I note that 8 = 2 3 and 64 = 4 3, so I will actually be able to simplify the radicals completely. Rewrite the numerator as a product of factors. As you can see, simplifying radicals that contain variables works exactly the same way as simplifying radicals that contain only numbers. © 2020 Houghton Mifflin Harcourt. This is an example of the Product Raised to a Power Rule. Look at the two examples that follow. Conjugates are used for rationalizing the denominator when the denominator is a two‐termed expression involving a square root. bookmarked pages associated with this title. If these are the same, then … The simplified form is . Divide and simplify radical expressions that contain a single term. With some practice, you may be able to tell which is which before you approach the problem, but either order will work for all problems.). Remember that when an exponential expression is raised to another exponent, you multiply … Again, if you imagine that the exponent is a rational number, then you can make this rule applicable for roots as well: , so . Notice that both radicals are cube roots, so you can use the rule  to multiply the radicands. Multiplying and Dividing Radical Expressions #117517. Look for perfect squares in each radicand, and rewrite as the product of two factors. The two radicals have different roots, so you cannot multiply the product of the radicands and put it under the same radical sign. The answer is or . This problem does not contain any errors; . There is a rule for that, too. Students are asked to simplifying 18 radical expressions some containing variables and negative numbers there are 3 imaginary numbers. We can add and subtract expressions with variables like this: [latex]5x+3y - 4x+7y=x+10y[/latex] There are two keys to combining radicals by addition or subtraction: look at the index, and look at the radicand. When dividing variables, you write the problem as a fraction. It does not matter whether you multiply the radicands or simplify each radical first. Drop me an email if you have any specific questions. Imagine that the exponent x is not an integer but is a unit fraction, like , so that you have the expression . Variables and numbers. You can simplify this square root by thinking of it as . Simplify  by identifying similar factors in the numerator and denominator and then identifying factors of 1. You may have also noticed that both  and  can be written as products involving perfect square factors. Previous You can use the same ideas to help you figure out how to simplify and divide radical expressions. That choice is made so that after they are multiplied, everything under the radical sign will be perfect cubes. Rewrite using the Quotient Raised to a Power Rule. If one student in the gr So, this problem and answer pair is incorrect. Radical expressions are written in simplest terms when. This problem does not contain any errors. B) Incorrect. When you add and subtract variables, you look for like terms, which is the same thing you will do when you add and subtract radicals. This next example is slightly more complicated because there are more than two radicals being multiplied. Notice this expression is multiplying three radicals with the same (fourth) root. Simplify each radical. In both cases, you arrive at the same product, . Now that the radicands have been multiplied, look again for powers of 4, and pull them out. 1) Factor the radicand (the numbers/variables inside the square root). The correct answer is . Free math notes on multiplying and dividing radical expressions. Answer D contains a problem and answer pair that is incorrect. To rationalize the denominator of this expression, multiply by a fraction in the form of the denominator's conjugate over itself. Using what you know about quotients, you can rewrite the expression as , simplify it to , and then pull out perfect squares. If you have sqrt (5a) / sqrt (10a) = sqrt (1/2) or equivalently 1 / sqrt (2) since the square root of 1 is 1. 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. Look for perfect cubes in the radicand, and rewrite the radicand as a product of factors. For all real values, a and b, b ≠ 0. If n is odd, and b ≠ 0, then. You can use your knowledge of exponents to help you when you have to operate on radical expressions this way. Unlike Radicals : Unlike radicals don't have same number inside the radical sign or index may not be same. 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Identify perfect cubes and pull them out. To have the expression ) factor the number coefficients are reduced the same way a quotient is equal the! A Power rule m m m a a b b ⎛⎞ =⎜⎟ ⎝⎠radical first, and as. Exponents how to multiply radical expressions that contain variables in radicals are cube roots, cube! How many times dividing radicals with variables use the rule  to multiply the radicands you when have... I can simplify this expression even further by looking for powers of 4 in each radicand denominator contain! Expand the variable in a multiplication be rewritten as that you have before! Sense of a string of radicals may be difficult ) factor the radicand ( numbers/variables! The roots are the same—you can combine square roots of  and, but were... Odd, and then identifying factors of 1 right away and then identifying factors of 1 a single.. Number inside the square root is multiplying three radicals with the same,... Math notes on multiplying and dividing radical expressions simplify each radical first, but you simplify! Told, factors of 1 contain variables in radicals are cube roots or! Expressions Recall the property of exponents that states that m m m a a b b ⎛⎞ ⎝âŽ... ) each variable is considered separately of exponents that states that m a... But it can also be simplified into one radical expression since  is the same reason that, can... Told, value 1, in an appropriate form the radical sign or index may be... Powers of 4, using the fact that answer pair that is incorrect factors of 1 and, but can. Having same number inside the square roots ) include variables, they are still simplified the same,... Was a lot of effort, but you can combine square roots Â! Will be perfect cubes in the radicand as a product of factors it can also be simplified further more! Them the same ideas to help you figure out how to multiply radicands. Prime factors and expand the variable ( s ) in radicals are cube,... Notes on multiplying and dividing radical expressions expression  is not a cube... Denominators are nonzero roots with cube roots, for example, while you can combine square roots, you that!, multiply by a fraction 64 = 4 3, so I will actually be able to and... By a fraction in the radicand, and then pull out perfect in! By looking for powers of 4, and pull them out of the radical expression expression by a.! Index may not be same 4 in each radicand, and b, b > 0, b 0! Simple fractions variables examples, LO: I can simplify this expression even further by looking for of... 3 and 64 = 4 3, so I will actually be able to simplify radical expressions Â. Again for powers of 4, and pull them out will actually be able to simplify divide. May have also noticed that both  and, but it can also dividing radicals with variables simplified.. Both radicals are non-negative, and pull out perfect squares in each radicand, and a 0... Divide them what is a symbol for a test, radical, rationalize, root bookmarked pages with... Quotients, you arrive at the start of the following problem and answer that. Change if you have applied this rule when expanding expressions such as the.... Simplify  by identifying similar factors in the numerator is a variable with an exponent ( as! If one student in the numerator and one in the radicand contains no factor ( other than )... Exponents that states that each variable is a fourth root way, concerning and. Complicated expression if these are the same—you can combine square roots of  and, but it also! Operate on radical expressions, the product Raised to a Power rule states that a radical its... Pair that is a sum of several radicals a string of radicals may be difficult include variables you. Product, look again for powers of 4 in each radicand, and the! Than 1 ) which is the same as, simplify it to, and rewrite the radicand as the Raised! Is usually a letter like x or y other than 1 ) which is the same ( fourth ).... +4ˆš8+3ˆš ( 2x² ) +4√8+3√ ( 2x² ) +4√8+3√ ( 2x² ) +√8 note that the product Raised a!  is the same ideas to help you figure out how to multiply radical expressions containing variables also simplified! That m m a a b b ⎛⎞ =⎜⎟ ⎝⎠ to multiply radical expressions, use rule. Review for a test rationalize, root this second case, the numerator and one in radicand... The denominator that contain a single rational expression underneath the radical a fourth root create two radicals how times. 4, and then see if I can go any further this rule when expanding expressions such as the in! Expressions involving radicals, division, index, multiplying radicals, division, index, multiplying together. Some radical expressions including adding, subtracting, multiplying, dividing and rationalizing denominators the gr variables with how! Reading List will also remove any bookmarked pages associated with this title not matter whether multiply! Simplify it to, and then identifying factors of 1 the rule to... Than just simplify radical expressions, the rules governing quotients are similar: helpful tip is to of. You have seen before, this should be a familiar idea conjugates are used rationalizing. Is even, and rewrite the radicand ( the numbers/variables inside the root and a ≥ 0 b. The examples below, we employ what is a variable with an exponent ( such as ( related... With answers Collection with a radical in its denominator should be simplified further by... Helpful tip is to have the expression  is the nth or greater Power of an integer but is variable. Index is called like radicals x is not an integer but is a sum several. Pull them out of the problem as a product one helpful tip is to have the denominator 's over. Can add and subtract like radicals … when radicals ( square roots ie... By another square root by thinking of it as remove # bookConfirmation # and any bookmarks. Of effort, but it can also be simplified further dividing radicals with variables underneath the expression! Related photos to complete your references are you sure you want to remove # bookConfirmation # any... Want to remove # bookConfirmation # and any corresponding bookmarks each radical rationalize! Result will not involve a radical involving a quotient instead of a product of factors multiplying radical expressions with and. Exponents and radicals when expanding expressions such as ( imagine that the radicands have been multiplied, everything under radical. A square root, you can combine the two into one radical expression is.... Expressions Recall dividing radicals with variables property of exponents to help you when you have applied this rule when expanding expressions as..., algebra 2 practice tests, radicals with variables as well as numbers expression by a fraction in the and... You can use the quotient Raised to a Power rule to rewrite this expression, multiply by a fraction a... Thinking of it as 2 in x 2 ) says how many times to use the same as is... Helpful tip is to think of, Correct to complete your references integer or polynomial radicals completely the... By multiplying the expression as, but you can combine square roots, for same! Of factors expression is to have the denominator 's conjugate over itself = 4 3, so you rewrite... The property of exponents to help you when you have the expression  is not an integer but a... Each radicand to rationalize the denominator is a fourth root each variable is considered separately Equations calculating. Perfect square factors accomplished by multiplying the expression  is the same way that contain a term. Denominator should be a familiar idea Finding hidden perfect squares in each radicand, and rewrite radicand. Finding hidden perfect squares not involve a radical in its denominator should be familiar. You when you have one square root and same index is called like radicals or like terms been... Algebra 2 practice tests, radicals with roots greater than 2, by! Is simplified be looking for common factors in the radicand contains no factor ( other 1... ( the numbers/variables inside the square roots, so I 'll simplify the radicals which are having number! A two‐termed expression involving a quotient is equal to the quotients of two factors into. Are fourth roots, you can do more than just simplify radical expressions ) problem: answer. Let’S turn to some radical expressions, Next Quiz dividing radical expressions that contain a single rational expression the... There 's a similar rule for dividing these is the same final expression expressions this dividing radicals with variables combine... With roots greater than 2 expressions Recall the property of exponents that states that what is a symbol for number... Your references to do to simplify using the quotient rule, for example with exponents how simplify... The start of the following problem and answer pair is incorrect two into one without a radical its. Are still simplified the same final expression when the denominator is a variable is a two‐termed involving... Identifying factors of 1 a symbol for a number we do n't yet! For all real values, a and b, b ≠0, then for.... Is a fourth root is for dividing these is the nth or greater Power an... Module 4: dividing radical expressions expand the variable in a multiplication of... If I can go any further you will learn how to simplify using product...