[latex] \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}[/latex]. [latex] \frac{\sqrt[3]{640}}{\sqrt[3]{40}}[/latex]. \\ & = 15 x \sqrt { 2 } - 5 \cdot 2 x \\ & = 15 x \sqrt { 2 } - 10 x \end{aligned}\). However, this is not the case for a cube root. \\ &= \frac { \sqrt { 20 } - \sqrt { 60 } } { 2 - 6 } \quad\quad\quad\quad\quad\quad\:\:\:\color{Cerulean}{Simplify.} \(\frac { x \sqrt { 2 } + 3 \sqrt { x y } + y \sqrt { 2 } } { 2 x - y }\), 49. \\ ( \sqrt { x } + \sqrt { y } ) ( \sqrt { x } - \sqrt { y } ) & = ( \sqrt { x } ) ^ { 2 } - ( \sqrt { y } ) ^ { 2 } \\ & = x - y \end{aligned}\), Multiply: \(( 3 - 2 \sqrt { y } ) ( 3 + 2 \sqrt { y } )\). \\ & = \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. In our first example, we will work with integers, and then we will move on to expressions with variable radicands. Apply the product rule for radicals, and then simplify. You multiply radical expressions that contain variables in the same manner. Given real numbers \(\sqrt [ n ] { A }\) and \(\sqrt [ n ] { B }\), \(\sqrt [ n ] { A } \cdot \sqrt [ n ] { B } = \sqrt [ n ] { A \cdot B }\)\. What if you found the quotient of this expression by dividing within the radical first and then took the cube root of the quotient? }\\ & = \sqrt [ 3 ] { 16 } \\ & = \sqrt [ 3 ] { 8 \cdot 2 } \color{Cerulean}{Simplify.} The answer is [latex]12{{x}^{3}}y,\,\,x\ge 0,\,\,y\ge 0[/latex]. The radius of the base of a right circular cone is given by \(r = \sqrt { \frac { 3 V } { \pi h } }\) where \(V\) represents the volume of the cone and \(h\) represents its height. \\ & = \frac { 2 x \sqrt [ 5 ] { 5 \cdot 2 ^ { 3 } x ^ { 2 } y ^ { 4 } } } { \sqrt [ 5 ] { 2 ^ { 5 } x ^ { 5 } y ^ { 5 } } } \quad\quad\:\:\color{Cerulean}{Simplify.} Look at the two examples that follow. \(\begin{aligned} \frac { \sqrt [ 3 ] { 2 } } { \sqrt [ 3 ] { 25 } } & = \frac { \sqrt [ 3 ] { 2 } } { \sqrt [ 3 ] { 5 ^ { 2 } } } \cdot \color{Cerulean}{\frac { \sqrt [ 3 ] { 5 } } { \sqrt [ 3 ] { 5 } } \:Multiply\:by\:the\:cube\:root\:of\:factors\:that\:result\:in\:powers\:of\:3.} The same is true of roots: [latex] \sqrt[x]{ab}=\sqrt[x]{a}\cdot \sqrt[x]{b}[/latex]. Dividing Radicals without Variables (Basic with no rationalizing). This website uses cookies to ensure you get the best experience. In both problems, the Product Raised to a Power Rule is used right away and then the expression is simplified. Multiplying Radical Expressions with Variables Using Distribution In all of these examples, multiplication of radicals has been shown following the pattern √a⋅√b =√ab a ⋅ b = a b. Simplify. A radical is a number or an expression under the root symbol. \\ & = \sqrt [ 3 ] { 72 } \quad\quad\:\color{Cerulean} { Simplify. } Identify factors of [latex]1[/latex], and simplify. }\\ & = 15 \sqrt { 2 x ^ { 2 } } - 5 \sqrt { 4 x ^ { 2 } } \quad\quad\quad\quad\:\:\:\color{Cerulean}{Simplify.} Next lesson. The answer is [latex]10{{x}^{2}}{{y}^{2}}\sqrt[3]{x}[/latex]. Look for perfect squares in the radicand. [latex] 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}[/latex], [latex] 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}[/latex]. Once we multiply the radicals, we then look for factors that are a power of the index and simplify the radical whenever possible. By multiplying the variable parts of the two radicals together, I'll get x 4 , which is the square of x 2 , so I'll be able to take x 2 out front, too. Type any radical equation into calculator , and the Math Way app will solve it form there. It is common practice to write radical expressions without radicals in the denominator. \(\frac { \sqrt [ 3 ] { 9 a b } } { 2 b }\), 21. Look at the two examples that follow. The product raised to a power rule that we discussed previously will help us find products of radical expressions. 19The process of determining an equivalent radical expression with a rational denominator. \(\frac { \sqrt [ 5 ] { 12 x y ^ { 3 } z ^ { 4 } } } { 2 y z }\), 29. The Product Raised to a Power Rule is important because you can use it to multiply radical expressions. [latex] 2\sqrt[4]{16{{x}^{9}}}\cdot \sqrt[4]{{{y}^{3}}}\cdot \sqrt[4]{81{{x}^{3}}y}[/latex], [latex] x\ge 0[/latex], [latex] y\ge 0[/latex]. Answers to Multiplying Radicals of Index 2: No Variable Factors 1) 6 2) 4 3) −8 6 4) 12 5) 36 10 6) 250 3 7) 3 2 + 2 15 8) 3 + 3 3 9) −25 5 − 5 15 10) 3 6 + 10 3 11) −10 5 − 5 2 12) −12 30 + 45 13) 1 14) 7 + 6 2 15) 8 − 4 3 16) −4 − 15 2 17) −34 + 2 10 18) −2 19) −32 + 5 6 20) 10 + 4 6 . \(\begin{aligned} \sqrt [ 3 ] { 12 } \cdot \sqrt [ 3 ] { 6 } & = \sqrt [ 3 ] { 12 \cdot 6 }\quad \color{Cerulean} { Multiply\: the\: radicands. } We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. \(\frac { 2 x + 1 + \sqrt { 2 x + 1 } } { 2 x }\), 53. 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