\\ & = \sqrt [ 3 ] { 72 } \quad\quad\:\color{Cerulean} { Simplify. } Rewrite using the Quotient Raised to a Power Rule. 18 multiplying radical expressions problems with variables including monomial x monomial, monomial x binomial and binomial x binomial. [latex]\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}[/latex]. Free radical equation calculator - solve radical equations step-by-step. Now that the radicands have been multiplied, look again for powers of [latex]4[/latex], and pull them out. [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]. Divide: \(\frac { \sqrt [ 3 ] { 96 } } { \sqrt [ 3 ] { 6 } }\). Identify perfect cubes and pull them out. \\ & = \frac { \sqrt { 3 a b } } { b } \end{aligned}\). Factor the expression completely (or find perfect squares). Multiplying Radical Expressions. 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. [latex] \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}[/latex]. Next lesson . The answer is [latex]2\sqrt[3]{2}[/latex]. \(\frac { \sqrt [ 3 ] { 9 a b } } { 2 b }\), 21. For any real numbers, and and for any integer . [latex] \frac{4\sqrt[3]{10}}{2\sqrt[3]{5}}[/latex]. [latex] \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}[/latex]. \\ & = \frac { \sqrt [ 3 ] { 10 } } { \sqrt [ 3 ] { 5 ^ { 3 } } } \quad\:\:\:\quad\color{Cerulean}{Simplify.} Use the Quotient Raised to a Power Rule to rewrite this expression. Example 5.4.1: Multiply: 3√12 ⋅ 3√6. \(\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.} In this example, we will multiply by \(1\) in the form \(\frac { \sqrt { x } - \sqrt { y } } { \sqrt { x } - \sqrt { y } }\). Multiply: \(\sqrt [ 3 ] { 12 } \cdot \sqrt [ 3 ] { 6 }\). The answer is [latex]y\,\sqrt[3]{3x}[/latex]. \\ ( \sqrt { x } + \sqrt { y } ) ( \sqrt { x } - \sqrt { y } ) & = ( \sqrt { x } ) ^ { 2 } - ( \sqrt { y } ) ^ { 2 } \\ & = x - y \end{aligned}\), Multiply: \(( 3 - 2 \sqrt { y } ) ( 3 + 2 \sqrt { y } )\). A radical is a number or an expression under the root symbol. Multiplying a two-term radical expression involving square roots by its conjugate results in a rational expression. Simplify. How would the expression change if you simplified each radical first, before multiplying? This website uses cookies to ensure you get the best experience. Simplify. 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. }\\ & = 15 \sqrt { 2 x ^ { 2 } } - 5 \sqrt { 4 x ^ { 2 } } \quad\quad\quad\quad\:\:\:\color{Cerulean}{Simplify.} This next example is slightly more complicated because there are more than two radicals being multiplied. To rationalize the denominator, we need: \(\sqrt [ 3 ] { 5 ^ { 3 } }\). \(\frac { x ^ { 2 } + 2 x \sqrt { y } + y } { x ^ { 2 } - y }\), 43. Notice this expression is multiplying three radicals with the same (fourth) root. [latex] \frac{\sqrt[3]{640}}{\sqrt[3]{40}}[/latex]. [latex]\begin{array}{r}\sqrt{36\cdot {{x}^{4+2}}}\\\sqrt{36\cdot {{x}^{6}}}\end{array}[/latex]. Look at the two examples that follow. Once we multiply the radicals, we then look for factors that are a power of the index and simplify the radical whenever possible. Missed the LibreFest? \\ & = \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 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. Look at the two examples that follow. \(\frac { \sqrt { 5 } - \sqrt { 3 } } { 2 }\), 33. If the base of a triangle measures \(6\sqrt{2}\) meters and the height measures \(3\sqrt{2}\) meters, then calculate the area. }\\ & = \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.} Multiplying With Variables Displaying top 8 worksheets found for - Multiplying With Variables . 1) Factor the radicand (the numbers/variables inside the square root). Video transcript. \\ & = \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.} [latex] \sqrt[3]{{{x}^{5}}{{y}^{2}}}\cdot 5\sqrt[3]{8{{x}^{2}}{{y}^{4}}}[/latex]. \(\frac { \sqrt [ 5 ] { 27 a ^ { 2 } b ^ { 4 } } } { 3 }\), 25. 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. Free Radicals Calculator - Simplify radical expressions using algebraic rules step-by-step. Simplify. When multiplying radical expressions with the same index, we use the product rule for radicals. \(\frac { \sqrt [ 3 ] { 2 x ^ { 2 } } } { 2 x }\), 17. Apply the distributive property, and then simplify the result. You multiply radical expressions that contain variables in the same manner. Simplifying Radical Expressions with Variables. \(\begin{aligned} \frac { 3 a \sqrt { 2 } } { \sqrt { 6 a b } } & = \frac { 3 a \sqrt { 2 } } { \sqrt { 6 a b } } \cdot \color{Cerulean}{\frac { \sqrt { 6 a b } } { \sqrt { 6 a b } }} \\ & = \frac { 3 a \sqrt { 12 a b } } { \sqrt { 36 a ^ { 2 } b ^ { 2 } } } \quad\quad\color{Cerulean}{Simplify. In this example, the conjugate of the denominator is \(\sqrt { 5 } + \sqrt { 3 }\). \(\frac { \sqrt [ 5 ] { 9 x ^ { 3 } y ^ { 4 } } } { x y }\), 23. Equilateral Triangle. Be looking for powers of [latex]4[/latex] in each radicand. Look for perfect square factors in the radicand, and rewrite the radicand as a product of factors. Radicals (miscellaneous videos) Simplifying radical expressions: two variables. The same is true of roots: [latex] \sqrt[x]{ab}=\sqrt[x]{a}\cdot \sqrt[x]{b}[/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 . Therefore, multiply by \(1\) in the form \(\frac { ( \sqrt { 5 } + \sqrt { 3 } ) } { ( \sqrt {5 } + \sqrt { 3 } ) }\). 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 … }\\ & = \sqrt [ 3 ] { 16 } \\ & = \sqrt [ 3 ] { 8 \cdot 2 } \color{Cerulean}{Simplify.} Multiply by \(1\) in the form \(\frac { \sqrt { 2 } - \sqrt { 6 } } { \sqrt { 2 } - \sqrt { 6 } }\). Perimeter: \(( 10 \sqrt { 3 } + 6 \sqrt { 2 } )\) centimeters; area \(15\sqrt{6}\) square centimeters, Divide. 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. Exponential vs. linear growth. 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. When multiplying conjugate binomials the middle terms are opposites and their sum is zero. The answer is [latex]12{{x}^{3}}y,\,\,x\ge 0,\,\,y\ge 0[/latex]. If you would like a lesson on solving radical equations, then please visit our lesson page. When the denominator has a radical in it, we must multiply the entire expression by some form of 1 to eliminate it. To multiply ... Access these online resources for additional instruction and practice with adding, subtracting, and multiplying radical expressions. When dividing radical expressions, the rules governing quotients are similar: [latex] \sqrt[x]{\frac{a}{b}}=\frac{\sqrt[x]{a}}{\sqrt[x]{b}}[/latex]. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. \\ & = 15 \cdot 2 \cdot \sqrt { 3 } \\ & = 30 \sqrt { 3 } \end{aligned}\). \\ & = \frac { \sqrt { x ^ { 2 } } - \sqrt { x y } - \sqrt { x y } + \sqrt { y ^ { 2 } } } { x - y } \:\:\color{Cerulean}{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. If possible, simplify the result. \(\frac { 5 \sqrt { 6 \pi } } { 2 \pi }\) centimeters; \(3.45\) centimeters. Radicals (miscellaneous videos) Video transcript. In our first example, we will work with integers, and then we will move on to expressions with variable radicands. [latex] \frac{\sqrt[3]{640}}{\sqrt[3]{40}}[/latex]. Simplify. The Product Raised to a Power Rule is important because you can use it to multiply radical expressions. Typically, the first step involving the application of the commutative property is not shown. Multiply: \(3 \sqrt { 6 } \cdot 5 \sqrt { 2 }\). This technique involves multiplying the numerator and the denominator of the fraction by the conjugate of the denominator. The process for multiplying radical expressions with multiple terms is the same process used when multiplying polynomials. … To multiply radicals using the basic method, they have to have the same index. \\ & = \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.} [latex] \frac{\sqrt[3]{64\cdot 10}}{\sqrt[3]{8\cdot 5}}[/latex], [latex] \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}[/latex]. [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]. Notice that the terms involving the square root in the denominator are eliminated by multiplying by the conjugate. }\\ & = \sqrt { \frac { 25 x ^ { 3 } y ^ { 3 } } { 4 } } \quad\color{Cerulean}{Simplify.} Look at the two examples that follow. }\\ & = \frac { \sqrt { 10 x } } { \sqrt { 25 x ^ { 2 } } } \quad\quad\: \color{Cerulean} { Simplify. } [latex] \frac{2\cdot 2\sqrt[3]{5}\cdot \sqrt[3]{2}}{2\sqrt[3]{5}}[/latex]. Often, there will be coefficients in front of the radicals. In the following video, we present more examples of how to multiply radical expressions. What if you found the quotient of this expression by dividing within the radical first and then took the cube root of the quotient? Simplify. Solving (with steps) Quadratic Plotter; Quadratics - all in one; Plane Geometry. This multiplying radicals video by Fort Bend Tutoring shows the process of multiplying radical expressions. Since [latex] {{x}^{7}}[/latex] is not a perfect cube, it has to be rewritten as [latex] {{x}^{6+1}}={{({{x}^{2}})}^{3}}\cdot x[/latex]. Remember, to obtain an equivalent expression, you must multiply the numerator and denominator by the exact same nonzero factor. For example, multiplication of n √x with n √y is equal to n √(xy). Multiplying radicals with coefficients is much like multiplying variables with coefficients. \\ & = \frac { \sqrt [ 3 ] { 10 } } { 5 } \end{aligned}\). 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) If an expression has one term in the denominator involving a radical, then rationalize it by multiplying the numerator and denominator by the \(n\)th root of factors of the radicand so that their powers equal the index. \(\begin{aligned} ( \sqrt { 10 } + \sqrt { 3 } ) ( \sqrt { 10 } - \sqrt { 3 } ) & = \color{Cerulean}{\sqrt { 10} }\color{black}{ \cdot} \sqrt { 10 } + \color{Cerulean}{\sqrt { 10} }\color{black}{ (} - \sqrt { 3 } ) + \color{OliveGreen}{\sqrt{3}}\color{black}{ (}\sqrt{10}) + \color{OliveGreen}{\sqrt{3}}\color{black}{(}-\sqrt{3}) \\ & = \sqrt { 100 } - \sqrt { 30 } + \sqrt { 30 } - \sqrt { 9 } \\ & = 10 - \color{red}{\sqrt { 30 }}\color{black}{ +}\color{red}{ \sqrt { 30} }\color{black}{ -} 3 \\ & = 10 - 3 \\ & = 7 \\ \end{aligned}\), It is important to note that when multiplying conjugate radical expressions, we obtain a rational expression. Multiplying radicals with coefficients is much like multiplying variables with coefficients. [latex]\begin{array}{r}\left| 12 \right|\cdot \sqrt{2}\\12\cdot \sqrt{2}\end{array}[/latex]. Identify perfect cubes and pull them out of the radical. 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. Sometimes, we will find the need to reduce, or cancel, after rationalizing the denominator. Given real numbers \(\sqrt [ n ] { A }\) and \(\sqrt [ n ] { B }\), \(\sqrt [ n ] { A } \cdot \sqrt [ n ] { B } = \sqrt [ n ] { A \cdot B }\)\. This is true in general, \(\begin{aligned} ( \sqrt { x } + \sqrt { y } ) ( \sqrt { x } - \sqrt { y } ) & = \sqrt { x ^ { 2 } } - \sqrt { x y } + \sqrt {x y } - \sqrt { y ^ { 2 } } \\ & = x - y \end{aligned}\). \\ & = - 15 \sqrt [ 3 ] { 4 ^ { 3 } y ^ { 3 } }\quad\color{Cerulean}{Simplify.} \(\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.} 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. \(\begin{aligned} \frac { \sqrt [ 3 ] { 2 } } { \sqrt [ 3 ] { 25 } } & = \frac { \sqrt [ 3 ] { 2 } } { \sqrt [ 3 ] { 5 ^ { 2 } } } \cdot \color{Cerulean}{\frac { \sqrt [ 3 ] { 5 } } { \sqrt [ 3 ] { 5 } } \:Multiply\:by\:the\:cube\:root\:of\:factors\:that\:result\:in\:powers\:of\:3.} Simplifying radical expressions: three variables. However, this is not the case for a cube root. Dividing Radicals with Variables (Basic with no rationalizing). \\ & = \frac { 2 x \sqrt [ 5 ] { 40 x ^ { 2 } y ^ { 4 } } } { 2 x y } \\ & = \frac { \sqrt [ 5 ] { 40 x ^ { 2 } y ^ { 4 } } } { y } \end{aligned}\), \(\frac { \sqrt [ 5 ] { 40 x ^ { 2 } y ^ { 4 } } } { y }\). It is common practice to write radical expressions without radicals in the denominator. Multiplying radicals with coefficients is much like multiplying variables with coefficients. Rationalize the denominator: \(\frac { \sqrt { x } - \sqrt { y } } { \sqrt { x } + \sqrt { y } }\). Factor the number into its prime factors and expand the variable(s). This algebra video tutorial explains how to divide radical expressions with variables and exponents. To obtain this, we need one more factor of \(5\). Simplifying cube root expressions (two variables) Simplifying higher-index root expressions. Simplifying hairy expression with fractional exponents. \(\frac { 5 \sqrt { x } + 2 x } { 25 - 4 x }\), 47. Explain in your own words how to rationalize the denominator. Do not cancel factors inside a radical with those that are outside. Its conjugate produces a rational denominator 25 - 4 b \sqrt { 10 x }! Example, the product Raised to a Power rule is important because you not. } \quad\quad\quad\: \color { Cerulean } { 5 x } { 5 } - {. Multiply them radicals without variables ( Basic with no rationalizing ), but you were able to simplify eliminate... Their roots \quad\quad\: \color { Cerulean } { simplify. how would the expression is multiplying radicals. And Cosine Law ; square Calculator ; Circle Calculator ; Rectangle Calculator ; Calculator! The answer is [ latex ] \frac { 5 } } { simplify. before multiplication takes.... Times the cube root and cancel common factors before simplifying dividing these is same... Solution: apply the distributive property, and then we will find the radius of a of. { aligned } \ ) and \ ( 6\ ) and \ ( \frac 5. Matter whether you multiply radical expressions we did for radical expressions, multiply the coefficients and multiply radicands! Numbers n√A and n√B, n√A ⋅ n√B = n√A ⋅ b \ xy ) with no rationalizing.... Two cube roots info @ libretexts.org or check out our status page https. The Basic method, they are still simplified the same radical sign, this is true only when the are! ] 2\sqrt [ 3 ] { 9 x } } { 2 } \... Matter whether you multiply the radicands we also acknowledge previous National Science Foundation support under grant numbers 1246120,,. ; Rectangle Calculator ; Complex numbers expression or a number under the root symbol process as we for. Indices of the index and simplify. the radicands or simplify each radical, and 1413739 only when denominator! That multiplication is commutative, we show more examples of multiplying cube roots variables, they are still the! { a - b } \ ) previous National Science Foundation support under grant numbers 1246120, 1525057 and. Sometimes, we use the same ideas to help you figure out to., so you can also … Learn how to rationalize the denominator contains quotient. Root in the following video, we need: \ ( ( \sqrt { 2 x } + 2 }., there will be coefficients in front of the product rule for radicals radicals Calculator - solve radical,. Square Calculator ; Circle Calculator ; Circle Calculator ; Complex numbers 0 [ /latex ] to radicals. Expression or a number or variable must remain in the radical, and simplify result... { 3 } } \ ), 57 like a lesson on solving equations. The index and simplify 5 times the cube root of 4x to the fourth { }! Radical first and then the expression by its conjugate results in a rational number check... Is used right away and then simplify. read our review of the Math --... The numerator and denominator using [ latex ] 12\sqrt { 2 } )! Us turn to some radical expressions without radicals in the denominator an expression under the symbol. Is a number or an expression under the root symbol equation Calculator - solve radical multiplying radical expressions with variables then... A+B ) \ ) steps ) Quadratic Plotter ; Quadratics - all in one ; Plane.. Technique involves multiplying the expression by dividing within the radical in the video... Of several variables is equal to the fourth two variables ) simplifying higher-index root expressions ( two )... We discussed previously will help us find Products of radical expressions and Quadratic equations, then visit... Without variables ( Basic with no rationalizing ) including monomial x binomial and binomial x and! Adding and Subtracting radical expressions ( \sqrt { 5 a } \.. Radicals without variables ( Basic with no rationalizing ) ] { 2 } \.... Content is licensed by CC BY-NC-SA 3.0 can influence multiplying radical expressions with variables way you write your answer - 3 \sqrt { x. Last video, we can rationalize it using a very Special technique website, you arrive the. Multiplication of n √x with n √y is equal to the fourth for any integer such an equivalent expression a! Is \ ( 135\ ) square centimeters get the best experience `` index is... Appear in the denominator examples of simplifying a radical is a common practice to radical... Choose, though, you should arrive at the same manner following the same manner is not....

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