The answer is $10{{x}^{2}}{{y}^{2}}\sqrt[3]{x}$. An expression with a radical in its denominator should be simplified into one without a radical in its denominator. In our next example, we will multiply two cube roots. http://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1/Preface, Use the product raised to a power rule to multiply radical expressions, Use the quotient raised to a power rule to divide radical expressions. When dividing radical expressions, the rules governing quotients are similar: $\sqrt[x]{\frac{a}{b}}=\frac{\sqrt[x]{a}}{\sqrt[x]{b}}$. We have a... We can divide the numerator and the denominator by y, so that would just become one. How would the expression change if you simplified each radical first, before multiplying? Simplify. $\sqrt{18}\cdot \sqrt{16}$. 4 is a factor, so we can split up the 24 as a 4 and a 6. In this tutorial we will be looking at rewriting and simplifying radical expressions. how to divide radical expressions; how to rationalize the denominator of a rational expression; Simplifying Radical Expressions Radical expressions are square roots of monomials, binomials, or polynomials. $\begin{array}{l}\sqrt[3]{\frac{8\cdot 3\cdot x\cdot {{y}^{3}}\cdot y}{8\cdot y}}\\\\\sqrt[3]{\frac{3\cdot x\cdot {{y}^{3}}}{1}\cdot \frac{8y}{8y}}\\\\\sqrt[3]{\frac{3\cdot x\cdot {{y}^{3}}}{1}\cdot 1}\end{array}$. Simplify. Simplify $\sqrt{\frac{30x}{10x}}$ by identifying similar factors in the numerator and denominator and then identifying factors of $1$. Simplifying radical expressions: three variables. Simplify. Multiply and divide radical expressions Use the product raised to a power rule to multiply radical expressions Use the quotient raised to a power rule to divide radical expressions You can do more than just simplify radical expressions. Whichever order you choose, though, you should arrive at the same final expression. The 6 doesn't have any factors that are perfect squares so the 6 will be left under the radical in the answer. Using the law of exponents, you divide the variables by subtracting the powers. Next look at the variable part. from your Reading List will also remove any For example, while you can think of $\frac{\sqrt{8{{y}^{2}}}}{\sqrt{225{{y}^{4}}}}$ as being equivalent to $\sqrt{\frac{8{{y}^{2}}}{225{{y}^{4}}}}$ since both the numerator and the denominator are square roots, notice that you cannot express $\frac{\sqrt{8{{y}^{2}}}}{\sqrt[4]{225{{y}^{4}}}}$ as $\sqrt[4]{\frac{8{{y}^{2}}}{225{{y}^{4}}}}$. Divide Radical Expressions. As you did with multiplication, you will start with some examples featuring integers before moving on to more complex expressions like $\frac{\sqrt[3]{24x{{y}^{4}}}}{\sqrt[3]{8y}}$. Even though our answer contained a variable with an odd exponent that was simplified from an even indexed root, we don’t need to write our answer with absolute value because we specified before we simplified that $x\ge 0$. Look for perfect squares in the radicand, and rewrite the radicand as the product of two factors. $\sqrt{{{(6)}^{2}}\cdot {{({{x}^{3}})}^{2}}}$, $\begin{array}{c}\sqrt{{{(6)}^{2}}}\cdot \sqrt{{{({{x}^{3}})}^{2}}}\\6\cdot {{x}^{3}}\end{array}$. $\sqrt[3]{{{x}^{5}}{{y}^{2}}}\cdot 5\sqrt[3]{8{{x}^{2}}{{y}^{4}}}$. Recall that the Product Raised to a Power Rule states that $\sqrt[x]{ab}=\sqrt[x]{a}\cdot \sqrt[x]{b}$. We give the Quotient Property of Radical Expressions again for easy reference. bookmarked pages associated with this title. ... Divide. Radical Expression Playlist on YouTube. $\begin{array}{c}\frac{\sqrt{16\cdot 3}}{\sqrt{25}}\\\\\text{or}\\\\\frac{\sqrt{4\cdot 4\cdot 3}}{\sqrt{5\cdot 5}}\end{array}$, $\begin{array}{r}\frac{\sqrt{{{(4)}^{2}}\cdot 3}}{\sqrt{{{(5)}^{2}}}}\\\\\frac{\sqrt{{{(4)}^{2}}}\cdot \sqrt{3}}{\sqrt{{{(5)}^{2}}}}\end{array}$, $\frac{4\cdot \sqrt{3}}{5}$. Well, what if you are dealing with a quotient instead of a product? Multiplying rational expressions. Divide radicals that have the same index number. There's a similar rule for dividing two radical expressions. The indices of the radicals must match in order to multiply them. 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. It is common practice to write radical expressions without radicals in the denominator. Use the rule $\sqrt[x]{\frac{a}{b}}=\frac{\sqrt[x]{a}}{\sqrt[x]{b}}$ to create two radicals; one in the numerator and one in the denominator. $\frac{\sqrt[3]{64\cdot 10}}{\sqrt[3]{8\cdot 5}}$, $\begin{array}{r}\frac{\sqrt[3]{{{(4)}^{3}}\cdot 10}}{\sqrt[3]{{{(2)}^{3}}\cdot 5}}\\\\\frac{\sqrt[3]{{{(4)}^{3}}}\cdot \sqrt[3]{10}}{\sqrt[3]{{{(2)}^{3}}}\cdot \sqrt[3]{5}}\\\\\frac{4\cdot \sqrt[3]{10}}{2\cdot \sqrt[3]{5}}\end{array}$. With some practice, you may be able to tell which is easier before you approach the problem, but either order will work for all problems. Let’s deal with them separately. $\sqrt{\frac{48}{25}}$. Note that we specify that the variable is non-negative, $x\ge 0$, thus allowing us to avoid the need for absolute value. Step 2: Simplify the coefficient. Since both radicals are cube roots, you can use the rule $\frac{\sqrt[x]{a}}{\sqrt[x]{b}}=\sqrt[x]{\frac{a}{b}}$ to create a single rational expression underneath the radical. Dividing Radical Expressions When dividing radical expressions, use the quotient rule. For all real values, a and b, b ≠ 0 If n is even, and a ≥ 0, b > 0, then Simplify. Radical expressions are written in simplest terms when. The Product Raised to a Power Rule is important because you can use it to multiply radical expressions. Now let us turn to some radical expressions containing division. Remember that when we multiply radicals with the same type of root, we just multiply the radicands and put the product under a radical sign. This calculator can be used to simplify a radical expression. Divide the coefficients, and divide the variables. $\sqrt[3]{\frac{640}{40}}$. Practice: Multiply & divide rational expressions (advanced) Next lesson. In our last video, we show more examples of simplifying radicals that contain quotients with variables. 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. A common way of dividing the radical expression is to have the denominator that contain no radicals. 2. You multiply radical expressions that contain variables in the same manner. $\begin{array}{r}640\div 40=16\\\sqrt[3]{16}\end{array}$. $\begin{array}{r}2\cdot 2\cdot 3\cdot {{x}^{2}}\cdot \sqrt[4]{x\cdot {{y}^{3}}\cdot {{x}^{3}}y}\\12{{x}^{2}}\sqrt[4]{{{x}^{1+3}}\cdot {{y}^{3+1}}}\end{array}$. That was a lot of effort, but you were able to simplify using the Quotient Raised to a Power Rule. In this second case, the numerator is a square root and the denominator is a fourth root. Use the quotient raised to a power rule to divide radical expressions (9.4.2) – Add and subtract radical expressions (9.4.3) – Multiply radicals with multiple terms (9.4.4) – Rationalize a denominator containing a radical expression How to divide algebraic terms or variables? 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. $\sqrt{{{(12)}^{2}}\cdot 2}$, $\sqrt{{{(12)}^{2}}}\cdot \sqrt{2}$. Recall that the Product Raised to a Power Rule states that $\sqrt[x]{ab}=\sqrt[x]{a}\cdot \sqrt[x]{b}$. Since ${{x}^{7}}$ is not a perfect cube, it has to be rewritten as ${{x}^{6+1}}={{({{x}^{2}})}^{3}}\cdot x$. 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 have used the Quotient Property of Radical Expressions to simplify roots of fractions. Dividing Radicals with Variables (Basic with no rationalizing). Rewrite using the Quotient Raised to a Power Rule. Simplify. Simplify each radical. $\sqrt[3]{3x{{y}^{3}}}\\\sqrt[3]{{{(y)}^{3}}\cdot \,3x}$, $\sqrt[3]{{{(y)}^{3}}}\cdot \,\sqrt[3]{3x}$. $\frac{\sqrt[3]{640}}{\sqrt[3]{40}}$. In both cases, you arrive at the same product, $12\sqrt{2}$. What if you found the quotient of this expression by dividing within the radical first and then took the cube root of the quotient? You can use the same ideas to help you figure out how to simplify and divide radical expressions. $\begin{array}{r}\sqrt{9\cdot 2}\cdot \sqrt{4\cdot 4}\\\sqrt{3\cdot 3\cdot 2}\cdot \sqrt{4\cdot 4}\end{array}$, $\sqrt{{{(3)}^{2}}\cdot 2}\cdot \sqrt{{{(4)}^{2}}}$, $\sqrt{{{(3)}^{2}}}\cdot \sqrt{2}\cdot \sqrt{{{(4)}^{2}}}$, $\begin{array}{c}\left|3\right|\cdot\sqrt{2}\cdot\left|4\right|\\3\cdot\sqrt{2}\cdot4\end{array}$. $\begin{array}{r}\left| 12 \right|\cdot \sqrt{2}\\12\cdot \sqrt{2}\end{array}$. The answer is $y\,\sqrt[3]{3x}$. You multiply radical expressions that contain variables in the same manner. For the numerical term 12, its largest perfect square factor is 4. Be looking for powers of $4$ in each radicand. Dividing Radical Expressions. Previous In the next video, we show more examples of simplifying a radical that contains a quotient. You can use the same ideas to help you figure out how to simplify and divide radical expressions. In the following video, we present more examples of how to multiply radical expressions. Notice that each group of numbers or variables gets written once when they move outside the radical because they are now one group. In the following video, we show more examples of multiplying cube roots. The multiplication of the denominator by its conjugate results in a whole number (okay, a negative, but the point is that there aren't any radicals): (Assume all variables are positive.) Rewrite the numerator as a product of factors. That choice is made so that after they are multiplied, everything under the radical sign will be perfect cubes. Within the radical, divide $640$ by $40$. Apply the distributive property when multiplying a radical expression with multiple terms. We give the Quotient Property of Radical Expressions again for easy reference. Multiplying a two-term radical expression involving square roots by its conjugate results in a rational expression. The conjugate of is . Look for perfect squares in each radicand, and rewrite as the product of two factors. Step 4: Simplify the expressions both inside and outside the radical by multiplying. Look for perfect cubes in the radicand, and rewrite the radicand as a product of factors. $2\sqrt[4]{{{(2)}^{4}}\cdot {{({{x}^{2}})}^{4}}\cdot x}\cdot \sqrt[4]{{{y}^{3}}}\cdot \sqrt[4]{{{(3)}^{4}}\cdot {{x}^{3}}y}$, $2\sqrt[4]{{{(2)}^{4}}}\cdot \sqrt[4]{{{({{x}^{2}})}^{4}}}\cdot \sqrt[4]{x}\cdot \sqrt[4]{{{y}^{3}}}\cdot \sqrt[4]{{{(3)}^{4}}}\cdot \sqrt[4]{{{x}^{3}}y}$. $\begin{array}{l}5\sqrt[3]{{{x}^{5}}{{y}^{2}}\cdot 8{{x}^{2}}{{y}^{4}}}\\5\sqrt[3]{8\cdot {{x}^{5}}\cdot {{x}^{2}}\cdot {{y}^{2}}\cdot {{y}^{4}}}\\5\sqrt[3]{8\cdot {{x}^{5+2}}\cdot {{y}^{2+4}}}\\5\sqrt[3]{8\cdot {{x}^{7}}\cdot {{y}^{6}}}\end{array}$. Well, what if you are dealing with a quotient instead of a product? Dividing radicals is really similar to multiplying radicals. A worked example of simplifying an expression that is a sum of several radicals. In the next example, we will use the same product from above to show that you can simplify before multiplying and get the same result. To rationalize the denominator of this expression, multiply by a fraction in the form of the denominator's conjugate over itself. You may have also noticed that both $\sqrt{18}$ and $\sqrt{16}$ can be written as products involving perfect square factors. We will need to use this property ‘in reverse’ to simplify a fraction with radicals. A perfect square is the … $\frac{\sqrt[3]{24x{{y}^{4}}}}{\sqrt[3]{8y}},\,\,y\ne 0$, $\sqrt[3]{\frac{24x{{y}^{4}}}{8y}}$. We will need to use this property ‘in reverse’ to simplify a fraction with radicals. One way of simplifying radical expressions is to break down the expression into perfect squares multiplying each other. Use the quotient rule to simplify radical expressions. Again, if you imagine that the exponent is a rational number, then you can make this rule applicable for roots as well: ${{\left( \frac{a}{b} \right)}^{\frac{1}{x}}}=\frac{{{a}^{\frac{1}{x}}}}{{{b}^{\frac{1}{x}}}}$, so $\sqrt[x]{\frac{a}{b}}=\frac{\sqrt[x]{a}}{\sqrt[x]{b}}$. If you have one square root divided by another square root, you can combine them together with division inside one square root. The same is true of roots: $\sqrt[x]{ab}=\sqrt[x]{a}\cdot \sqrt[x]{b}$. • The radicand and the index must be the same in order to add or subtract radicals. $\begin{array}{r}\sqrt[3]{{{(2)}^{3}}\cdot 2}\\\sqrt[3]{{(2)}^{3}}\cdot\sqrt[3]{2}\end{array}$. 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. 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. Simplify. $\frac{\sqrt[3]{640}}{\sqrt[3]{40}}$. When dividing radical expressions, use the quotient rule. There is a rule for that, too. Simplifying hairy expression with fractional exponents. Look for perfect cubes in the radicand, and rewrite the radicand as a product of factors. Multiply all numbers and variables inside the radical together. This property can be used to combine two radicals into one. Conjugates are used for rationalizing the denominator when the denominator is a two‐termed expression involving a square root. In both problems, the Product Raised to a Power Rule is used right away and then the expression is simplified. What can be multiplied with so the result will not involve a radical? Quiz Multiplying Radical Expressions, Next $\begin{array}{r}2\cdot \frac{2\sqrt[3]{5}}{2\sqrt[3]{5}}\cdot \sqrt[3]{2}\\\\2\cdot 1\cdot \sqrt[3]{2}\end{array}$. $\begin{array}{l}12{{x}^{2}}\sqrt[4]{{{x}^{4}}\cdot {{y}^{4}}}\\12{{x}^{2}}\sqrt[4]{{{x}^{4}}}\cdot \sqrt[4]{{{y}^{4}}}\\12{{x}^{2}}\cdot \left| x \right|\cdot \left| y \right|\end{array}$. So when you divide one radical expression by another, you can simplify it by writing both expressions under the same radical, then simplifying. Simplify. Dividing Radicals without Variables (Basic with no rationalizing). 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. We can only take the square root of variables with an EVEN power (the square root of x … 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. Example 7: Simplify the radical expression \sqrt {12{x^2}{y^4}} . We can divide an algebraic term by another algebraic term to get the quotient. $\begin{array}{r}\sqrt{36\cdot {{x}^{4+2}}}\\\sqrt{36\cdot {{x}^{6}}}\end{array}$. We can drop the absolute value signs in our final answer because at the start of the problem we were told $x\ge 0$, $y\ge 0$. 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). Dividing Algebraic Expressions . Step 1: Write the division of the algebraic terms as a fraction. $\begin{array}{r}\sqrt{18\cdot 16}\\\sqrt{288}\end{array}$. Perfect Powers 1 Simplify any radical expressions that are perfect squares. © 2020 Houghton Mifflin Harcourt. Even the smallest statement like $x\ge 0$ can influence the way you write your answer. Identify perfect cubes and pull them out. The steps below show how the division is carried out. Simplify, using $\sqrt{{{x}^{2}}}=\left| x \right|$. Welcome to MathPortal. 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. 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. Multiplying and Dividing Radical Expressions As long as the indices are the same, we can multiply the radicands together using the following property. Since all the radicals are fourth roots, you can use the rule $\sqrt[x]{ab}=\sqrt[x]{a}\cdot \sqrt[x]{b}$ to multiply the radicands. There is a rule for that, too. It is important to read the problem very well when you are doing math. 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. The quotient rule works only if: 1. Now take another look at that problem using this approach. Recall the rule: For any numbers a and b and any integer x: ${{(ab)}^{x}}={{a}^{x}}\cdot {{b}^{x}}$, For any numbers a and b and any positive integer x: ${{(ab)}^{\frac{1}{x}}}={{a}^{\frac{1}{x}}}\cdot {{b}^{\frac{1}{x}}}$, For any numbers a and b and any positive integer x: $\sqrt[x]{ab}=\sqrt[x]{a}\cdot \sqrt[x]{b}$. The answer is or . The denominator here contains a radical, but that radical is part of a larger expression. Look for perfect square factors in the radicand, and rewrite the radicand as a product of factors. Look for perfect cubes in the radicand. Dividing rational expressions: unknown expression. Recall that ${{x}^{4}}\cdot x^2={{x}^{4+2}}$. It can also be used the other way around to split a radical into two if there's a fraction inside. Removing #book# In our first example, we will work with integers, and then we will move on to expressions with variable radicands. $\frac{4\sqrt[3]{10}}{2\sqrt[3]{5}}$. The radicand contains both numbers and variables. Identify factors of $1$, and simplify. Use the Quotient Raised to a Power Rule to rewrite this expression. Use the quotient rule to divide radical expressions. $\begin{array}{r}2\cdot \left| 2 \right|\cdot \left| {{x}^{2}} \right|\cdot \sqrt[4]{x}\cdot \sqrt[4]{{{y}^{3}}}\cdot \left| 3 \right|\cdot \sqrt[4]{{{x}^{3}}y}\\2\cdot 2\cdot {{x}^{2}}\cdot \sqrt[4]{x}\cdot \sqrt[4]{{{y}^{3}}}\cdot 3\cdot \sqrt[4]{{{x}^{3}}y}\end{array}$. For any real numbers a and b (b ≠ 0) and any positive integer x: ${{\left( \frac{a}{b} \right)}^{\frac{1}{x}}}=\frac{{{a}^{\frac{1}{x}}}}{{{b}^{\frac{1}{x}}}}$, For any real numbers a and b (b ≠ 0) and any positive integer x: $\sqrt[x]{\frac{a}{b}}=\frac{\sqrt[x]{a}}{\sqrt[x]{b}}$. The quotient of the radicals is equal to the radical of the quotient. $5\sqrt[3]{{{(2)}^{3}}\cdot {{({{x}^{2}})}^{3}}\cdot x\cdot {{({{y}^{2}})}^{3}}}$, $\begin{array}{r}5\sqrt[3]{{{(2)}^{3}}}\cdot \sqrt[3]{{{({{x}^{2}})}^{3}}}\cdot \sqrt[3]{{{({{y}^{2}})}^{3}}}\cdot \sqrt[3]{x}\\5\cdot 2\cdot {{x}^{2}}\cdot {{y}^{2}}\cdot \sqrt[3]{x}\end{array}$. 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. Identify and pull out powers of $4$, using the fact that $\sqrt[4]{{{x}^{4}}}=\left| x \right|$. Divide Radical Expressions. Simplifying radical expressions: two variables. The answer is $\frac{4\sqrt{3}}{5}$. It does not matter whether you multiply the radicands or simplify each radical first. Simplify. By using this website, you agree to our Cookie Policy. Then simplify and combine all like radicals. Simplify each radical, if possible, before multiplying. Look at the two examples that follow. 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