Multiply: \(( \sqrt { x } - 5 \sqrt { y } ) ^ { 2 }\). \\ & = 15 \cdot \sqrt { 12 } \quad\quad\quad\:\color{Cerulean}{Multiply\:the\:coefficients\:and\:the\:radicands.} \\ & = 15 \sqrt { 4 \cdot 3 } \quad\quad\quad\:\color{Cerulean}{Simplify.} Research and discuss some of the reasons why it is a common practice to rationalize the denominator. Dividing Radicals without Variables (Basic with no rationalizing). Rationalize the denominator: \(\frac { \sqrt { 2 } } { \sqrt { 5 x } }\). Give the exact answer and the approximate answer rounded to the nearest hundredth. In the next video, we show more examples of simplifying a radical that contains a quotient. If we apply the quotient rule for radicals and write it as a single cube root, we will be able to reduce the fractional radicand. \(\begin{aligned} 3 \sqrt { 6 } \cdot 5 \sqrt { 2 } & = \color{Cerulean}{3 \cdot 5}\color{black}{ \cdot}\color{OliveGreen}{ \sqrt { 6 } \cdot \sqrt { 2} }\quad\color{Cerulean}{Multiplication\:is\:commutative.} Typically, the first step involving the application of the commutative property is not shown. Given real numbers n√A and n√B, n√A ⋅ n√B = n√A ⋅ B \. \(\sqrt { 6 } + \sqrt { 14 } - \sqrt { 15 } - \sqrt { 35 }\), 49. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. The answer is [latex]y\,\sqrt[3]{3x}[/latex]. Missed the LibreFest? In both problems, the Product Raised to a Power Rule is used right away and then the expression is simplified. To divide radical expressions with the same index, we use the quotient rule for radicals. Look for perfect squares in the radicand, and rewrite the radicand as the product of two factors. The radius of a sphere is given by \(r = \sqrt [ 3 ] { \frac { 3 V } { 4 \pi } }\) where \(V\) represents the volume of the sphere. Recall the rule: For any numbers a and b and any integer x: [latex] {{(ab)}^{x}}={{a}^{x}}\cdot {{b}^{x}}[/latex], For any numbers a and b and any positive integer x: [latex] {{(ab)}^{\frac{1}{x}}}={{a}^{\frac{1}{x}}}\cdot {{b}^{\frac{1}{x}}}[/latex], For any numbers a and b and any positive integer x: [latex] \sqrt[x]{ab}=\sqrt[x]{a}\cdot \sqrt[x]{b}[/latex]. If the base of a triangle measures \(6\sqrt{3}\) meters and the height measures \(3\sqrt{6}\) meters, then calculate the area. The radicand in the denominator determines the factors that you need to use to rationalize it. Factor the number into its prime factors and expand the variable(s). Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor. \(\begin{aligned} \frac { \sqrt { 2 } } { \sqrt { 5 x } } & = \frac { \sqrt { 2 } } { \sqrt { 5 x } } \cdot \color{Cerulean}{\frac { \sqrt { 5 x } } { \sqrt { 5 x } } { \:Multiply\:by\: } \frac { \sqrt { 5 x } } { \sqrt { 5 x } } . In both problems, the Product Raised to a Power Rule is used right away and then the expression is simplified. Rewrite the numerator as a product of factors. This mean that, the root of the product of several variables is equal to the product of their roots. In this case, we can see that \(6\) and \(96\) have common factors. Note that we specify that the variable is non-negative, [latex] x\ge 0[/latex], thus allowing us to avoid the need for absolute value. [latex] \sqrt{\frac{48}{25}}[/latex]. To multiply ... subtracting, and multiplying radical expressions. and ; Spec Look at the two examples that follow. Now that the radicands have been multiplied, look again for powers of [latex]4[/latex], and pull them out. Simplify, using [latex] \sqrt{{{x}^{2}}}=\left| x \right|[/latex]. Apply the distributive property when multiplying a radical expression with multiple terms. Notice this expression is multiplying three radicals with the same (fourth) root. 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. 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 [latex] x\ge 0[/latex]. For every pair of a number or variable under the radical, they become one when simplified. What is the perimeter and area of a rectangle with length measuring \(5\sqrt{3}\) centimeters and width measuring \(3\sqrt{2}\) centimeters? Explain in your own words how to rationalize the denominator. To multiply radicals using the basic method, they have to have the same index. The answer is [latex]\frac{4\sqrt{3}}{5}[/latex]. \\ ( \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 { 3 a \sqrt { 4 \cdot 3 a b} } { 6 ab } \\ & = \frac { 6 a \sqrt { 3 a b } } { b }\quad\quad\:\:\color{Cerulean}{Cancel.} (Assume all variables represent non-negative real numbers. Given real numbers \(\sqrt [ n ] { A }\) and \(\sqrt [ n ] { B }\), \(\frac { \sqrt [ n ] { A } } { \sqrt [ n ] { B } } = \sqrt [n]{ \frac { A } { B } }\). [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]. Simplify. [latex] \sqrt[3]{3x{{y}^{3}}}\\\sqrt[3]{{{(y)}^{3}}\cdot \,3x}[/latex], [latex] \sqrt[3]{{{(y)}^{3}}}\cdot \,\sqrt[3]{3x}[/latex]. \(\begin{aligned} \sqrt [ 3 ] { \frac { 27 a } { 2 b ^ { 2 } } } & = \frac { \sqrt [ 3 ] { 3 ^ { 3 } a } } { \sqrt [ 3 ] { 2 b ^ { 2 } } } \quad\quad\quad\quad\color{Cerulean}{Apply\:the\:quotient\:rule\:for\:radicals.} Adding and Subtracting Radical Expressions Quiz: Adding and Subtracting Radical Expressions What Are Radicals? Next lesson . For example, \(\frac { 1 } { \sqrt [ 3 ] { x } } \cdot \color{Cerulean}{\frac { \sqrt [ 3 ] { x } } { \sqrt [ 3 ] { x } }}\color{black}{ =} \frac { \sqrt [ 3 ] { x } } { \sqrt [ 3 ] { x ^ { 2 } } }\). Simplifying hairy expression with fractional exponents. 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. Since both radicals are cube roots, you can use the rule [latex] \frac{\sqrt[x]{a}}{\sqrt[x]{b}}=\sqrt[x]{\frac{a}{b}}[/latex] to create a single rational expression underneath the radical. When multiplying radical expressions with the same index, we use the product rule for radicals. The Product Raised to a Power Rule is important because you can use it to multiply radical expressions. Simplify. That was a lot of effort, but you were able to simplify using the Quotient Raised to a Power Rule. Simplify [latex] \sqrt{\frac{30x}{10x}}[/latex] by identifying similar factors in the numerator and denominator and then identifying factors of [latex]1[/latex]. [latex] \frac{\sqrt[3]{640}}{\sqrt[3]{40}}[/latex]. Identify and pull out powers of [latex]4[/latex], using the fact that [latex] \sqrt[4]{{{x}^{4}}}=\left| x \right|[/latex]. It is common practice to write radical expressions without radicals in the denominator. (Assume all variables represent positive real numbers. ), Rationalize the denominator. Multiplying Radical Expressions. What is the perimeter and area of a rectangle with length measuring \(2\sqrt{6}\) centimeters and width measuring \(\sqrt{3}\) centimeters? Find the radius of a sphere with volume \(135\) square centimeters. Equilateral Triangle. 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. \(\frac { \sqrt [ 5 ] { 12 x y ^ { 3 } z ^ { 4 } } } { 2 y z }\), 29. \(\frac { 1 } { \sqrt [ 3 ] { x } } = \frac { 1 } { \sqrt [ 3 ] { x } } \cdot \color{Cerulean}{\frac { \sqrt [ 3 ] { x ^ { 2 } } } { \sqrt [ 3 ] { x ^ { 2 } } }} = \frac { \sqrt [ 3 ] { x ^ { 2 } } } { \sqrt [ 3 ] { x ^ { 3 } } } = \frac { \sqrt [ 3 ] { x ^ { 2 } } } { x }\). [latex] \sqrt{{{(6)}^{2}}\cdot {{({{x}^{3}})}^{2}}}[/latex], [latex] \begin{array}{c}\sqrt{{{(6)}^{2}}}\cdot \sqrt{{{({{x}^{3}})}^{2}}}\\6\cdot {{x}^{3}}\end{array}[/latex]. 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 contains plenty of examples and practice problems. Recall that [latex] {{x}^{4}}\cdot x^2={{x}^{4+2}}[/latex]. Multiply and simplify 5 times the cube root of 2x squared times 3 times the cube root of 4x to the fourth. Multiply: \(\sqrt [ 3 ] { 6 x ^ { 2 } y } \left( \sqrt [ 3 ] { 9 x ^ { 2 } y ^ { 2 } } - 5 \cdot \sqrt [ 3 ] { 4 x y } \right)\). \(\frac { - 5 - 3 \sqrt { 5 } } { 2 }\), 37. The "index" is the very small number written just to the left of the uppermost line in the radical symbol. [latex] \sqrt{{{(12)}^{2}}\cdot 2}[/latex], [latex] \sqrt{{{(12)}^{2}}}\cdot \sqrt{2}[/latex]. Notice that the process for dividing these is the same as it is for dividing integers. If a pair does not exist, the number or variable must remain in the radicand. A radical is an expression or a number under the root symbol. In the following video, we show more examples of multiplying cube roots. \\ & = \frac { \sqrt { 25 x ^ { 3 } y ^ { 3 } } } { \sqrt { 4 } } \\ & = \frac { 5 x y \sqrt { x y } } { 2 } \end{aligned}\). \\ & = - 15 \sqrt [ 3 ] { 4 ^ { 3 } y ^ { 3 } }\quad\color{Cerulean}{Simplify.} When multiplying conjugate binomials the middle terms are opposites and their sum is zero. 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. For example, while you can think of [latex] \frac{\sqrt{8{{y}^{2}}}}{\sqrt{225{{y}^{4}}}}[/latex] as being equivalent to [latex] \sqrt{\frac{8{{y}^{2}}}{225{{y}^{4}}}}[/latex] since both the numerator and the denominator are square roots, notice that you cannot express [latex] \frac{\sqrt{8{{y}^{2}}}}{\sqrt[4]{225{{y}^{4}}}}[/latex] as [latex] \sqrt[4]{\frac{8{{y}^{2}}}{225{{y}^{4}}}}[/latex]. It is common practice to write radical expressions without radicals in the denominator. To rationalize the denominator, we need: \(\sqrt [ 3 ] { 5 ^ { 3 } }\). However, this is not the case for a cube root. [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]. In this case, if we multiply by \(1\) in the form of \(\frac { \sqrt [ 3 ] { x ^ { 2 } } } { \sqrt [ 3 ] { x ^ { 2 } } }\), then we can write the radicand in the denominator as a power of \(3\). 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