# multiplying radical expressions with variables

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Product Property of Roots . The product raised to a power rule that we discussed previously will help us find products of radical expressions. Legal. Use the distributive property when multiplying rational expressions with more than one term. Simplify. Simplifying radical expressions: three variables. Use the rule $\sqrt[x]{a}\cdot \sqrt[x]{b}=\sqrt[x]{ab}$ to multiply the radicands. $\sqrt[3]{{{x}^{5}}{{y}^{2}}}\cdot 5\sqrt[3]{8{{x}^{2}}{{y}^{4}}}$. Apply the distributive property, and then combine like terms. Be looking for powers of $4$ in each radicand. \\ &= \frac { \sqrt { 4 \cdot 5 } - \sqrt { 4 \cdot 15 } } { - 4 } \\ &= \frac { 2 \sqrt { 5 } - 2 \sqrt { 15 } } { - 4 } \\ &=\frac{2(\sqrt{5}-\sqrt{15})}{-4} \\ &= \frac { \sqrt { 5 } - \sqrt { 15 } } { - 2 } = - \frac { \sqrt { 5 } - \sqrt { 15 } } { 2 } = \frac { - \sqrt { 5 } + \sqrt { 15 } } { 2 } \end{aligned}\), $$\frac { \sqrt { 15 } - \sqrt { 5 } } { 2 }$$. Recall that ${{x}^{4}}\cdot x^2={{x}^{4+2}}$. Adding and Subtracting Radical Expressions Quiz: Adding and Subtracting Radical Expressions What Are Radicals? $$\frac { - 5 - 3 \sqrt { 5 } } { 2 }$$, 37. In both problems, the Product Raised to a Power Rule is used right away and then the expression is simplified. Apply the distributive property when multiplying a radical expression with multiple terms. Simplifying hairy expression with fractional exponents. $\sqrt[3]{\frac{640}{40}}$. Identify perfect cubes and pull them out of the radical. The process of finding such an equivalent expression is called rationalizing the denominator. Look for perfect cubes in the radicand. $$\frac { \sqrt [ 3 ] { 6 } } { 3 }$$, 15. $$\frac { \sqrt [ 5 ] { 9 x ^ { 3 } y ^ { 4 } } } { x y }$$, 23. Simplify. Use the Quotient Raised to a Power Rule to rewrite this expression. The answer is $10{{x}^{2}}{{y}^{2}}\sqrt[3]{x}$. Look for perfect squares in the radicand. In this example, multiply by $$1$$ in the form $$\frac { \sqrt { 5 x } } { \sqrt { 5 x } }$$. $\frac{\sqrt{48}}{\sqrt{25}}$. Rationalize the denominator: $$\frac { \sqrt { 10 } } { \sqrt { 2 } + \sqrt { 6 } }$$. Recall that the Product Raised to a Power Rule states that $\sqrt[x]{ab}=\sqrt[x]{a}\cdot \sqrt[x]{b}$. Identify perfect cubes and pull them out. When multiplying radical expressions with the same index, we use the product rule for radicals. $5\sqrt[3]{{{(2)}^{3}}\cdot {{({{x}^{2}})}^{3}}\cdot x\cdot {{({{y}^{2}})}^{3}}}$, $\begin{array}{r}5\sqrt[3]{{{(2)}^{3}}}\cdot \sqrt[3]{{{({{x}^{2}})}^{3}}}\cdot \sqrt[3]{{{({{y}^{2}})}^{3}}}\cdot \sqrt[3]{x}\\5\cdot 2\cdot {{x}^{2}}\cdot {{y}^{2}}\cdot \sqrt[3]{x}\end{array}$. \\ & = 15 \cdot \sqrt { 12 } \quad\quad\quad\:\color{Cerulean}{Multiply\:the\:coefficients\:and\:the\:radicands.} Apply the product rule for radicals, and then simplify. The "index" is the very small number written just to the left of the uppermost line in the radical symbol. In both problems, the Product Raised to a Power Rule is used right away and then the expression is simplified. Simplify. In this lesson, we are only going to deal with square roots only which is a specific type of radical expression with an index of \color{red}2.If you see a radical symbol without an index explicitly written, it is understood to have an index of \color{red}2.. Below are the basic rules in multiplying radical expressions. Rationalize the denominator: Multiply numerator and denominator by the 5th root of of factors that will result in 5th powers of each factor in the radicand of the denominator. }\\ & = \sqrt { \frac { 25 x ^ { 3 } y ^ { 3 } } { 4 } } \quad\color{Cerulean}{Simplify.} You can multiply and divide them, too. An expression with a radical in its denominator should be simplified into one without a radical in its denominator. This mean that, the root of the product of several variables is equal to the product of their roots. Do not cancel factors inside a radical with those that are outside. Notice that $$b$$ does not cancel in this example. Simplify $\sqrt{\frac{30x}{10x}}$ by identifying similar factors in the numerator and denominator and then identifying factors of $1$. Learn more Accept. 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. To do this, multiply the fraction by a special form of $$1$$ so that the radicand in the denominator can be written with a power that matches the index. Since ${{x}^{7}}$ is not a perfect cube, it has to be rewritten as ${{x}^{6+1}}={{({{x}^{2}})}^{3}}\cdot x$. }\\ & = 15 \sqrt { 2 x ^ { 2 } } - 5 \sqrt { 4 x ^ { 2 } } \quad\quad\quad\quad\:\:\:\color{Cerulean}{Simplify.} Rationalize the denominator: $$\sqrt { \frac { 9 x } { 2 y } }$$. Free Radicals Calculator - Simplify radical expressions using algebraic rules step-by-step. $\begin{array}{r}\sqrt{18\cdot 16}\\\sqrt{288}\end{array}$. $$\frac { 5 \sqrt { 6 \pi } } { 2 \pi }$$ centimeters; $$3.45$$ centimeters. Right Triangle; Sine and Cosine Law ; Square Calculator; Rectangle Calculator; Circle Calculator; Complex Numbers. You can do more than just simplify radical expressions. \\ & = \frac { x - 2 \sqrt { x y } + y } { x - y } \end{aligned}\), $$\frac { x - 2 \sqrt { x y } + y } { x - y }$$, Rationalize the denominator: $$\frac { 2 \sqrt { 3 } } { 5 - \sqrt { 3 } }$$, Multiply. 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. Learn how to multiply radicals. Multiplying a two-term radical expression involving square roots by its conjugate results in a rational expression. 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: $\sqrt[x]{ab}=\sqrt[x]{a}\cdot \sqrt[x]{b}$. }\\ & = \frac { 3 \sqrt [ 3 ] { 4 a b } } { 2 b } \end{aligned}\), $$\frac { 3 \sqrt [ 3 ] { 4 a b } } { 2 b }$$, Rationalize the denominator: $$\frac { 2 x \sqrt [ 5 ] { 5 } } { \sqrt [ 5 ] { 4 x ^ { 3 } y } }$$, In this example, we will multiply by $$1$$ in the form $$\frac { \sqrt [ 5 ] { 2 ^ { 3 } x ^ { 2 } y ^ { 4 } } } { \sqrt [ 5 ] { 2 ^ { 3 } x ^ { 2 } y ^ { 4 } } }$$, \begin{aligned} \frac{2x\sqrt[5]{5}}{\sqrt[5]{4x^{3}y}} & = \frac{2x\sqrt[5]{5}}{\sqrt[5]{2^{2}x^{3}y}}\cdot\color{Cerulean}{\frac{\sqrt[5]{2^{3}x^{2}y^{4}}}{\sqrt[5]{2^{3}x^{2}y^{4}}} \:\:Multiply\:by\:the\:fifth\:root\:of\:factors\:that\:result\:in\:pairs.} Note that we specify that the variable is non … $\begin{array}{r}2\cdot \frac{2\sqrt[3]{5}}{2\sqrt[3]{5}}\cdot \sqrt[3]{2}\\\\2\cdot 1\cdot \sqrt[3]{2}\end{array}$. Notice how much more straightforward the approach was. \(\frac { a - 2 \sqrt { a b + b } } { a - b }, 45. By using this website, you agree to our Cookie Policy. How would the expression change if you simplified each radical first, before multiplying? For any real numbers, and and for any integer . (Assume all variables represent non-negative real numbers. Look at the two examples that follow. \\ & = \frac { 3 \sqrt [ 3 ] { a } } { \sqrt [ 3 ] { 2 b ^ { 2 } } } \cdot \color{Cerulean}{\frac { \sqrt [ 3 ] { 2 ^ { 2 } b } } { \sqrt [ 3 ] { 2 ^ { 2 } b } }\:\:\:Multiply\:by\:the\:cube\:root\:of\:factors\:that\:result\:in\:powers.} It advisable to place factor in the same radical sign, this is possible when the variables are simplified to a common index. 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. $$\frac { \sqrt { 5 } - \sqrt { 3 } } { 2 }$$, 33. Multiplying Radical Expressions. When multiplying conjugate binomials the middle terms are opposites and their sum is zero. … This technique involves multiplying the numerator and the denominator of the fraction by the conjugate of the denominator. Simplify. This multiplying radicals video by Fort Bend Tutoring shows the process of multiplying radical expressions. When the denominator (divisor) of a radical expression contains a radical, it is a common practice to find an equivalent expression where the denominator is a rational number. \\ & = \frac { 3 \sqrt [ 3 ] { 2 ^ { 2 } ab } } { \sqrt [ 3 ] { 2 ^ { 3 } b ^ { 3 } } } \quad\quad\quad\color{Cerulean}{Simplify. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. Remember, to obtain an equivalent expression, you must multiply the numerator and denominator by the exact same nonzero factor. In the following video, we present more examples of how to multiply radical expressions. Apply the distributive property and multiply each term by $$5 \sqrt { 2 x }$$. }\\ & = \frac { \sqrt { 10 x } } { \sqrt { 25 x ^ { 2 } } } \quad\quad\: \color{Cerulean} { Simplify. } Look at the two examples that follow. Notice that both radicals are cube roots, so you can use the rule  to multiply the radicands. Factor the number into its prime factors and expand the variable(s). \begin{aligned} \frac{\sqrt{10}}{\sqrt{2}+\sqrt{6} }&= \frac{(\sqrt{10})}{(\sqrt{2}+\sqrt{6})} \color{Cerulean}{\frac{(\sqrt{2}-\sqrt{6})}{(\sqrt{2}-\sqrt{6})}\quad\quad Multiple\:by\:the\:conjugate.} We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Solution: Apply the product rule for radicals, and then simplify. \\ & = \frac { \sqrt [ 3 ] { 10 } } { 5 } \end{aligned}. In this example, we will multiply by $$1$$ in the form $$\frac { \sqrt { x } - \sqrt { y } } { \sqrt { x } - \sqrt { y } }$$. Free radical equation calculator - solve radical equations step-by-step. To rationalize the denominator, we need: $$\sqrt [ 3 ] { 5 ^ { 3 } }$$. Multiply by $$1$$ in the form $$\frac { \sqrt { 2 } - \sqrt { 6 } } { \sqrt { 2 } - \sqrt { 6 } }$$. $\sqrt{\frac{48}{25}}$. This process is called rationalizing the denominator. 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. Simplify. You can use the same ideas to help you figure out how to simplify and divide radical expressions. In this example, we will multiply by $$1$$ in the form $$\frac { \sqrt { 6 a b } } { \sqrt { 6 a b } }$$. Missed the LibreFest? $$\frac { \sqrt { 75 } } { \sqrt { 3 } }$$, $$\frac { \sqrt { 360 } } { \sqrt { 10 } }$$, $$\frac { \sqrt { 72 } } { \sqrt { 75 } }$$, $$\frac { \sqrt { 90 } } { \sqrt { 98 } }$$, $$\frac { \sqrt { 90 x ^ { 5 } } } { \sqrt { 2 x } }$$, $$\frac { \sqrt { 96 y ^ { 3 } } } { \sqrt { 3 y } }$$, $$\frac { \sqrt { 162 x ^ { 7 } y ^ { 5 } } } { \sqrt { 2 x y } }$$, $$\frac { \sqrt { 363 x ^ { 4 } y ^ { 9 } } } { \sqrt { 3 x y } }$$, $$\frac { \sqrt [ 3 ] { 16 a ^ { 5 } b ^ { 2 } } } { \sqrt [ 3 ] { 2 a ^ { 2 } b ^ { 2 } } }$$, $$\frac { \sqrt [ 3 ] { 192 a ^ { 2 } b ^ { 7 } } } { \sqrt [ 3 ] { 2 a ^ { 2 } b ^ { 2 } } }$$, $$\frac { \sqrt { 2 } } { \sqrt { 3 } }$$, $$\frac { \sqrt { 3 } } { \sqrt { 7 } }$$, $$\frac { \sqrt { 3 } - \sqrt { 5 } } { \sqrt { 3 } }$$, $$\frac { \sqrt { 6 } - \sqrt { 2 } } { \sqrt { 2 } }$$, $$\frac { 3 b ^ { 2 } } { 2 \sqrt { 3 a b } }$$, $$\frac { 1 } { \sqrt [ 3 ] { 3 y ^ { 2 } } }$$, $$\frac { 9 x \sqrt[3] { 2 } } { \sqrt [ 3 ] { 9 x y ^ { 2 } } }$$, $$\frac { 5 y ^ { 2 } \sqrt [ 3 ] { x } } { \sqrt [ 3 ] { 5 x ^ { 2 } y } }$$, $$\frac { 3 a } { 2 \sqrt [ 3 ] { 3 a ^ { 2 } b ^ { 2 } } }$$, $$\frac { 25 n } { 3 \sqrt [ 3 ] { 25 m ^ { 2 } n } }$$, $$\frac { 3 } { \sqrt [ 5 ] { 27 x ^ { 2 } y } }$$, $$\frac { 2 } { \sqrt [ 5 ] { 16 x y ^ { 2 } } }$$, $$\frac { a b } { \sqrt [ 5 ] { 9 a ^ { 3 } b } }$$, $$\frac { a b c } { \sqrt [ 5 ] { a b ^ { 2 } c ^ { 3 } } }$$, $$\sqrt [ 5 ] { \frac { 3 x } { 8 y ^ { 2 } z } }$$, $$\sqrt [ 5 ] { \frac { 4 x y ^ { 2 } } { 9 x ^ { 3 } y z ^ { 4 } } }$$, $$\frac { 1 } { \sqrt { 5 } + \sqrt { 3 } }$$, $$\frac { 1 } { \sqrt { 7 } - \sqrt { 2 } }$$, $$\frac { \sqrt { 3 } } { \sqrt { 3 } + \sqrt { 6 } }$$, $$\frac { \sqrt { 5 } } { \sqrt { 5 } + \sqrt { 15 } }$$, $$\frac { - 2 \sqrt { 2 } } { 4 - 3 \sqrt { 2 } }$$, $$\frac { \sqrt { 3 } + \sqrt { 5 } } { \sqrt { 3 } - \sqrt { 5 } }$$, $$\frac { \sqrt { 10 } - \sqrt { 2 } } { \sqrt { 10 } + \sqrt { 2 } }$$, $$\frac { 2 \sqrt { 3 } - 3 \sqrt { 2 } } { 4 \sqrt { 3 } + \sqrt { 2 } }$$, $$\frac { 6 \sqrt { 5 } + 2 } { 2 \sqrt { 5 } - \sqrt { 2 } }$$, $$\frac { x - y } { \sqrt { x } + \sqrt { y } }$$, $$\frac { x - y } { \sqrt { x } - \sqrt { y } }$$, $$\frac { x + \sqrt { y } } { x - \sqrt { y } }$$, $$\frac { x - \sqrt { y } } { x + \sqrt { y } }$$, $$\frac { \sqrt { a } - \sqrt { b } } { \sqrt { a } + \sqrt { b } }$$, $$\frac { \sqrt { a b } + \sqrt { 2 } } { \sqrt { a b } - \sqrt { 2 } }$$, $$\frac { \sqrt { x } } { 5 - 2 \sqrt { x } }$$, $$\frac { \sqrt { x } + \sqrt { 2 y } } { \sqrt { 2 x } - \sqrt { y } }$$, $$\frac { \sqrt { 3 x } - \sqrt { y } } { \sqrt { x } + \sqrt { 3 y } }$$, $$\frac { \sqrt { 2 x + 1 } } { \sqrt { 2 x + 1 } - 1 }$$, $$\frac { \sqrt { x + 1 } } { 1 - \sqrt { x + 1 } }$$, $$\frac { \sqrt { x + 1 } + \sqrt { x - 1 } } { \sqrt { x + 1 } - \sqrt { x - 1 } }$$, $$\frac { \sqrt { 2 x + 3 } - \sqrt { 2 x - 3 } } { \sqrt { 2 x + 3 } + \sqrt { 2 x - 3 } }$$. \\ & = 15 \cdot 2 \cdot \sqrt { 3 } \\ & = 30 \sqrt { 3 } \end{aligned}\). Multiply: $$3 \sqrt { 6 } \cdot 5 \sqrt { 2 }$$. Multiply: $$\sqrt [ 3 ] { 6 x ^ { 2 } y } \left( \sqrt [ 3 ] { 9 x ^ { 2 } y ^ { 2 } } - 5 \cdot \sqrt [ 3 ] { 4 x y } \right)$$. Multiplying Radical Expressions To multiply radical expressions (square roots)... 1) Multiply the numbers/variables outside the radicand (square root) 2) Multiply the numbers/variables inside the radicand (square root) Identify and pull out powers of $4$, using the fact that $\sqrt[4]{{{x}^{4}}}=\left| x \right|$. Now that the radicands have been multiplied, look again for powers of $4$, and pull them out. $\frac{\sqrt{30x}}{\sqrt{10x}},x>0$. \\ & = 15 x \sqrt { 2 } - 5 \cdot 2 x \\ & = 15 x \sqrt { 2 } - 10 x \end{aligned}\). $$\frac { 1 } { \sqrt [ 3 ] { x } } = \frac { 1 } { \sqrt [ 3 ] { x } } \cdot \color{Cerulean}{\frac { \sqrt [ 3 ] { x ^ { 2 } } } { \sqrt [ 3 ] { x ^ { 2 } } }} = \frac { \sqrt [ 3 ] { x ^ { 2 } } } { \sqrt [ 3 ] { x ^ { 3 } } } = \frac { \sqrt [ 3 ] { x ^ { 2 } } } { x }$$. \\ & = 2 \sqrt [ 3 ] { 2 } \end{aligned}\). Multiplying radicals with coefficients is much like multiplying variables with coefficients. In our first example, we will work with integers, and then we will move on to expressions with variable radicands. $\sqrt{12{{x}^{4}}}\cdot \sqrt{3x^2}$, $x\ge 0$, $\sqrt{12{{x}^{4}}\cdot 3x^2}\\\sqrt{12\cdot 3\cdot {{x}^{4}}\cdot x^2}$. $$3 \sqrt [ 3 ] { 2 } - 2 \sqrt [ 3 ] { 15 }$$, 47. Identify factors of $1$, and simplify. Multiplying radicals with coefficients is much like multiplying variables with coefficients. The goal is to find an equivalent expression without a radical in the denominator. \\ & = 15 \sqrt { 4 \cdot 3 } \quad\quad\quad\:\color{Cerulean}{Simplify.} \\ & = \frac { \sqrt { 5 } + \sqrt { 3 } } { 5-3 } \\ & = \frac { \sqrt { 5 } + \sqrt { 3 } } { 2 } \end{aligned}\), $$\frac { \sqrt { 5 } + \sqrt { 3 } } { 2 }$$. You multiply radical expressions that contain variables in the same manner. Apply the distributive property, simplify each radical, and then combine like terms. 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. \begin{aligned} \frac { \sqrt { x } - \sqrt { y } } { \sqrt { x } + \sqrt { y } } & = \frac { ( \sqrt { x } - \sqrt { y } ) } { ( \sqrt { x } + \sqrt { y } ) } \color{Cerulean}{\frac { ( \sqrt { x } - \sqrt { y } ) } { ( \sqrt { x } - \sqrt { y } ) } \quad \quad Multiply\:by\:the\:conjugate\:of\:the\:denominator.} As you did with multiplication, you will start with some examples featuring integers before moving on to more complex expressions like $\frac{\sqrt[3]{24x{{y}^{4}}}}{\sqrt[3]{8y}}$. Aptitute test paper, solve algebra problems free, crossword puzzle in trigometry w/answer, linear algebra points of a parabola, Mathamatics for kids, right triangles worksheets for 3rd … The indices of the radicals must match in order to multiply them. Sometimes, we will find the need to reduce, or cancel, after rationalizing the denominator. $\begin{array}{l}5\sqrt[3]{{{x}^{5}}{{y}^{2}}\cdot 8{{x}^{2}}{{y}^{4}}}\\5\sqrt[3]{8\cdot {{x}^{5}}\cdot {{x}^{2}}\cdot {{y}^{2}}\cdot {{y}^{4}}}\\5\sqrt[3]{8\cdot {{x}^{5+2}}\cdot {{y}^{2+4}}}\\5\sqrt[3]{8\cdot {{x}^{7}}\cdot {{y}^{6}}}\end{array}$. When the denominator has a radical in it, we must multiply the entire expression by some form of 1 to eliminate it. $\begin{array}{r}\left| 12 \right|\cdot \sqrt{2}\\12\cdot \sqrt{2}\end{array}$. It contains plenty of examples and practice problems. The same is true of roots: $\sqrt[x]{ab}=\sqrt[x]{a}\cdot \sqrt[x]{b}$. 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