# how to divide radical expressions with variables

In our last video, we show more examples of simplifying radicals that contain quotients with variables. In the following video, we present more examples of how to multiply radical expressions. $\sqrt{{{(6)}^{2}}\cdot {{({{x}^{3}})}^{2}}}$, $\begin{array}{c}\sqrt{{{(6)}^{2}}}\cdot \sqrt{{{({{x}^{3}})}^{2}}}\\6\cdot {{x}^{3}}\end{array}$. Look for perfect square factors in the radicand, and rewrite the radicand as a product of factors. What if you found the quotient of this expression by dividing within the radical first and then took the cube root of the quotient? 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. Divide radicals that have the same index number. So when you divide one radical expression by another, you can simplify it by writing both expressions under the same radical, then simplifying. $\frac{4\sqrt{10}}{2\sqrt{5}}$. $\begin{array}{r}\sqrt{{{(2)}^{3}}\cdot 2}\\\sqrt{{(2)}^{3}}\cdot\sqrt{2}\end{array}$. Whichever order you choose, though, you should arrive at the same final expression. Simplify. 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. and any corresponding bookmarks? $\frac{2\cdot 2\sqrt{5}\cdot \sqrt{2}}{2\sqrt{5}}$. That choice is made so that after they are multiplied, everything under the radical sign will be perfect cubes. Simplify each radical, if possible, before multiplying. bookmarked pages associated with this title. $\begin{array}{r}\sqrt{9\cdot 2}\cdot \sqrt{4\cdot 4}\\\sqrt{3\cdot 3\cdot 2}\cdot \sqrt{4\cdot 4}\end{array}$, $\sqrt{{{(3)}^{2}}\cdot 2}\cdot \sqrt{{{(4)}^{2}}}$, $\sqrt{{{(3)}^{2}}}\cdot \sqrt{2}\cdot \sqrt{{{(4)}^{2}}}$, $\begin{array}{c}\left|3\right|\cdot\sqrt{2}\cdot\left|4\right|\\3\cdot\sqrt{2}\cdot4\end{array}$. You can do more than just simplify radical expressions. Apply the distributive property when multiplying a radical expression with multiple terms. There is a rule for that, too. $\begin{array}{r}2\cdot 2\cdot 3\cdot {{x}^{2}}\cdot \sqrt{x\cdot {{y}^{3}}\cdot {{x}^{3}}y}\\12{{x}^{2}}\sqrt{{{x}^{1+3}}\cdot {{y}^{3+1}}}\end{array}$. If you have one square root divided by another square root, you can combine them together with division inside one square root. Dividing radicals is really similar to multiplying radicals. Identify perfect cubes and pull them out. Notice how much more straightforward the approach was. 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. The quotient rule works only if: 1. Identify factors of $1$, and simplify. 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 6 doesn't have any factors that are perfect squares so the 6 will be left under the radical in the answer. Previous Simplify. Multiplying rational expressions: multiple variables. 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. Dividing rational expressions: unknown expression. We will need to use this property ‘in reverse’ to simplify a fraction with radicals. $\begin{array}{r}2\cdot \frac{2\sqrt{5}}{2\sqrt{5}}\cdot \sqrt{2}\\\\2\cdot 1\cdot \sqrt{2}\end{array}$. 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. 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. 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. Quiz Multiplying Radical Expressions, Next Divide the coefficients, and divide the variables. We can divide, we have y minus two divided by y minus two, so those cancel out. We can divide an algebraic term by another algebraic term to get the quotient. Simplify. 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. You multiply radical expressions that contain variables in the same manner. Are you sure you want to remove #bookConfirmation# Use the quotient rule to divide radical expressions. Be looking for powers of $4$ in each radicand. Simplify. We can only take the square root of variables with an EVEN power (the square root of x … $\frac{\sqrt{640}}{\sqrt{40}}$. Rationalizing the Denominator. Simplify. To get rid of it, I'll multiply by the conjugate in order to "simplify" this expression. A worked example of simplifying an expression that is a sum of several radicals. This property can be used to combine two radicals into one. The answer is or . • Sometimes it is necessary to simplify radicals first to find out if they can be added 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. Dividing Radical Expressions. Multiply all numbers and variables inside the radical together. Notice that both radicals are cube roots, so you can use the rule  to multiply the radicands. There is a rule for that, too. Simplify. In the next video, we show more examples of simplifying a radical that contains a quotient. $\begin{array}{r}\left| 12 \right|\cdot \sqrt{2}\\12\cdot \sqrt{2}\end{array}$. The answer is $\frac{4\sqrt{3}}{5}$. The indices of the radicals must match in order to multiply them. In our first example, we will work with integers, and then we will move on to expressions with variable radicands. 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. In both cases, you arrive at the same product, $12\sqrt{2}$. $\begin{array}{l}12{{x}^{2}}\sqrt{{{x}^{4}}\cdot {{y}^{4}}}\\12{{x}^{2}}\sqrt{{{x}^{4}}}\cdot \sqrt{{{y}^{4}}}\\12{{x}^{2}}\cdot \left| x \right|\cdot \left| y \right|\end{array}$. Divide Radical Expressions. We have used the Quotient Property of Radical Expressions to simplify roots of fractions. $\begin{array}{r}\sqrt{18\cdot 16}\\\sqrt{288}\end{array}$. Dividing Radical Expressions When dividing radical expressions, use the quotient rule. Simplifying hairy expression with fractional exponents. Well, what if you are dealing with a quotient instead of a product? As you did with multiplication, you will start with some examples featuring integers before moving on to more complex expressions like $\frac{\sqrt{24x{{y}^{4}}}}{\sqrt{8y}}$. https://www.khanacademy.org/.../v/multiply-and-simplify-a-radical-expression-2 Unit 16: Radical Expressions and Quadratic Equations, from Developmental Math: An Open Program. To rationalize the denominator of this expression, multiply by a fraction in the form of the denominator's conjugate over itself. Now take another look at that problem using this approach. $\frac{\sqrt{48}}{\sqrt{25}}$. 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. One way of simplifying radical expressions is to break down the expression into perfect squares multiplying each other. Look for perfect cubes in the radicand, and rewrite the radicand as a product of factors. This algebra video tutorial shows you how to perform many operations to simplify radical expressions. Within the radical, divide $640$ by $40$. Step 2: Simplify the coefficient. Note that you cannot multiply a square root and a cube root using this rule. $\sqrt{18}\cdot \sqrt{16}$. A perfect square is the … Now let's see. Adding and subtracting rational expressions intro. $\begin{array}{r}2\cdot \left| 2 \right|\cdot \left| {{x}^{2}} \right|\cdot \sqrt{x}\cdot \sqrt{{{y}^{3}}}\cdot \left| 3 \right|\cdot \sqrt{{{x}^{3}}y}\\2\cdot 2\cdot {{x}^{2}}\cdot \sqrt{x}\cdot \sqrt{{{y}^{3}}}\cdot 3\cdot \sqrt{{{x}^{3}}y}\end{array}$. Simplify $\sqrt{\frac{24x{{y}^{4}}}{8y}}$ by identifying similar factors in the numerator and denominator and then identifying factors of $1$. Now let's think about it. 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). Multiply all numbers and variables outside the radical together. $\sqrt{{{x}^{5}}{{y}^{2}}}\cdot 5\sqrt{8{{x}^{2}}{{y}^{4}}}$. The quotient of the radicals is equal to the radical of the quotient. $\sqrt{12{{x}^{4}}}\cdot \sqrt{3x^2}$, $x\ge 0$, $\sqrt{12{{x}^{4}}\cdot 3x^2}\\\sqrt{12\cdot 3\cdot {{x}^{4}}\cdot x^2}$. You can simplify this expression even further by looking for common factors in the numerator and denominator. $\sqrt{\frac{48}{25}}$. We will need to use this property ‘in reverse’ to simplify a fraction with radicals. Sort by: Top Voted. Simplify $\sqrt{\frac{30x}{10x}}$ by identifying similar factors in the numerator and denominator and then identifying factors of $1$. 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 $\frac{\sqrt{8{{y}^{2}}}}{\sqrt{225{{y}^{4}}}}$ as $\sqrt{\frac{8{{y}^{2}}}{225{{y}^{4}}}}$. The answer is $2\sqrt{2}$. Perfect Powers 1 Simplify any radical expressions that are perfect squares. It is important to read the problem very well when you are doing math. $\sqrt{\frac{640}{40}}$. 2. Next look at the variable part. Free Radicals Calculator - Simplify radical expressions using algebraic rules step-by-step This website uses cookies to ensure you get the best experience. Step 1: Write the division of the algebraic terms as a fraction. The radicand contains no factor (other than 1) which is the nth or greater power of an integer or polynomial. This web site owner is mathematician Miloš Petrović. Quiz Dividing Radical Expressions. This is accomplished by multiplying the expression by a fraction having the value 1, in an appropriate form. When dividing radical expressions, the rules governing quotients are similar: $\sqrt[x]{\frac{a}{b}}=\frac{\sqrt[x]{a}}{\sqrt[x]{b}}$. Simplifying radical expressions: three variables. $5\sqrt{{{(2)}^{3}}\cdot {{({{x}^{2}})}^{3}}\cdot x\cdot {{({{y}^{2}})}^{3}}}$, $\begin{array}{r}5\sqrt{{{(2)}^{3}}}\cdot \sqrt{{{({{x}^{2}})}^{3}}}\cdot \sqrt{{{({{y}^{2}})}^{3}}}\cdot \sqrt{x}\\5\cdot 2\cdot {{x}^{2}}\cdot {{y}^{2}}\cdot \sqrt{x}\end{array}$. Well, what if you are dealing with a quotient instead of a product? We have used the Quotient Property of Radical Expressions to simplify roots of fractions. Assume that the variables are positive. Simplify. Recall that ${{x}^{4}}\cdot x^2={{x}^{4+2}}$. Look for perfect squares in the radicand. The answer is $12{{x}^{3}}y,\,\,x\ge 0,\,\,y\ge 0$. 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