\\ & = \frac { \sqrt [ 3 ] { 10 } } { \sqrt [ 3 ] { 5 ^ { 3 } } } \quad\:\:\:\quad\color{Cerulean}{Simplify.} Note that we specify that the variable is non-negative, $x\ge 0$, thus allowing us to avoid the need for absolute value. In both problems, the Product Raised to a Power Rule is used right away and then the expression is simplified. $\frac{\sqrt[3]{640}}{\sqrt[3]{40}}$. $$\sqrt { 6 } + \sqrt { 14 } - \sqrt { 15 } - \sqrt { 35 }$$, 49. You multiply radical expressions that contain variables in the same manner. The answer is $12{{x}^{3}}y,\,\,x\ge 0,\,\,y\ge 0$. In this example, the conjugate of the denominator is $$\sqrt { 5 } + \sqrt { 3 }$$. This website uses cookies to ensure you get the best experience. \\ & = \frac { \sqrt { 25 x ^ { 3 } y ^ { 3 } } } { \sqrt { 4 } } \\ & = \frac { 5 x y \sqrt { x y } } { 2 } \end{aligned}\). $$2 a \sqrt { 7 b } - 4 b \sqrt { 5 a }$$, 45. $$\frac { a - 2 \sqrt { a b + b } } { a - b }$$, 45. $\sqrt{12{{x}^{4}}}\cdot \sqrt{3x^2}$, $x\ge 0$, $\sqrt{12{{x}^{4}}\cdot 3x^2}\\\sqrt{12\cdot 3\cdot {{x}^{4}}\cdot x^2}$. To multiply ... subtracting, and multiplying radical expressions. For any real numbers, and and for any integer . $\begin{array}{r}\sqrt{36\cdot {{x}^{4+2}}}\\\sqrt{36\cdot {{x}^{6}}}\end{array}$. Multiply: $$\sqrt [ 3 ] { 12 } \cdot \sqrt [ 3 ] { 6 }$$. 19The process of determining an equivalent radical expression with a rational denominator. \\ & = \frac { 2 x \sqrt [ 5 ] { 40 x ^ { 2 } y ^ { 4 } } } { 2 x y } \\ & = \frac { \sqrt [ 5 ] { 40 x ^ { 2 } y ^ { 4 } } } { y } \end{aligned}\), $$\frac { \sqrt [ 5 ] { 40 x ^ { 2 } y ^ { 4 } } } { y }$$. The goal is to find an equivalent expression without a radical in the denominator. You can multiply and divide them, too. \\ & = \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. Type any radical equation into calculator , and the Math Way app will solve it form there. You can do more than just simplify radical expressions. Now that the radicands have been multiplied, look again for powers of $4$, and pull them out. $\sqrt{18}\cdot \sqrt{16}$. Rationalize the denominator: $$\sqrt { \frac { 9 x } { 2 y } }$$. 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. Rationalize the denominator: $$\frac { \sqrt [ 3 ] { 2 } } { \sqrt [ 3 ] { 25 } }$$. Do not cancel factors inside a radical with those that are outside. (Assume $$y$$ is positive.). Often, there will be coefficients in front of the radicals. Recall that the Product Raised to a Power Rule states that $\sqrt[x]{ab}=\sqrt[x]{a}\cdot \sqrt[x]{b}$. When two terms involving square roots appear in the denominator, we can rationalize it using a very special technique. Learn more Accept. The product raised to a power rule that we discussed previously will help us find products of radical expressions. \begin{aligned} \frac { \sqrt [ 3 ] { 96 } } { \sqrt [ 3 ] { 6 } } & = \sqrt [ 3 ] { \frac { 96 } { 6 } } \quad\color{Cerulean}{Apply\:the\:quotient\:rule\:for\:radicals\:and\:reduce\:the\:radicand. $\frac{\sqrt{48}}{\sqrt{25}}$. Simplifying radical expressions: three variables. 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. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. Look for perfect squares in the radicand. Apply the distributive property, simplify each radical, and then combine like terms. 18The factors \((a+b) and $$(a-b)$$ are conjugates. Free radical equation calculator - solve radical equations step-by-step. Rewrite the numerator as a product of factors. Give the exact answer and the approximate answer rounded to the nearest hundredth. Apply the distributive property when multiplying a radical expression with multiple terms. $$18 \sqrt { 2 } + 2 \sqrt { 3 } - 12 \sqrt { 6 } - 4$$, 57. \\ & = 15 x \sqrt { 2 } - 5 \cdot 2 x \\ & = 15 x \sqrt { 2 } - 10 x \end{aligned}\). $$\frac { x \sqrt { 2 } + 3 \sqrt { x y } + y \sqrt { 2 } } { 2 x - y }$$, 49. $\begin{array}{r}2\cdot \left| 2 \right|\cdot \left| {{x}^{2}} \right|\cdot \sqrt[4]{x}\cdot \sqrt[4]{{{y}^{3}}}\cdot \left| 3 \right|\cdot \sqrt[4]{{{x}^{3}}y}\\2\cdot 2\cdot {{x}^{2}}\cdot \sqrt[4]{x}\cdot \sqrt[4]{{{y}^{3}}}\cdot 3\cdot \sqrt[4]{{{x}^{3}}y}\end{array}$. $$\frac { 3 \sqrt [ 3 ] { 6 x ^ { 2 } y } } { y }$$, 19. $$\frac { x ^ { 2 } + 2 x \sqrt { y } + y } { x ^ { 2 } - y }$$, 43. Give the exact answer and the approximate answer rounded to the nearest hundredth. Look at the two examples that follow. \\ & = \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.} You can simplify this expression even further by looking for common factors in the numerator and denominator. $\sqrt{\frac{48}{25}}$. Simplifying cube root expressions (two variables) Simplifying higher-index root expressions. Given real numbers n√A and n√B, n√A ⋅ n√B = n√A ⋅ B \. Simplify. 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. The result is $$12xy$$. There is a rule for that, too. The process of finding such an equivalent expression is called rationalizing the denominator. Simplifying hairy expression with fractional exponents. It is important to read the problem very well when you are doing math. In this example, we will multiply by $$1$$ in the form $$\frac { \sqrt { x } - \sqrt { y } } { \sqrt { x } - \sqrt { y } }$$. You multiply radical expressions that contain variables in the same manner. With some practice, you may be able to tell which is easier before you approach the problem, but either order will work for all problems. It advisable to place factor in the same radical sign, this is possible when the variables are simplified to a common index. This mean that, the root of the product of several variables is equal to the product of their roots. Radical Expressions. Multiply: $$( \sqrt { 10 } + \sqrt { 3 } ) ( \sqrt { 10 } - \sqrt { 3 } )$$. Then, only after multiplying, some radicals have been simplified—like in the last problem. $$\frac { \sqrt { 75 } } { \sqrt { 3 } }$$, $$\frac { \sqrt { 360 } } { \sqrt { 10 } }$$, $$\frac { \sqrt { 72 } } { \sqrt { 75 } }$$, $$\frac { \sqrt { 90 } } { \sqrt { 98 } }$$, $$\frac { \sqrt { 90 x ^ { 5 } } } { \sqrt { 2 x } }$$, $$\frac { \sqrt { 96 y ^ { 3 } } } { \sqrt { 3 y } }$$, $$\frac { \sqrt { 162 x ^ { 7 } y ^ { 5 } } } { \sqrt { 2 x y } }$$, $$\frac { \sqrt { 363 x ^ { 4 } y ^ { 9 } } } { \sqrt { 3 x y } }$$, $$\frac { \sqrt [ 3 ] { 16 a ^ { 5 } b ^ { 2 } } } { \sqrt [ 3 ] { 2 a ^ { 2 } b ^ { 2 } } }$$, $$\frac { \sqrt [ 3 ] { 192 a ^ { 2 } b ^ { 7 } } } { \sqrt [ 3 ] { 2 a ^ { 2 } b ^ { 2 } } }$$, $$\frac { \sqrt { 2 } } { \sqrt { 3 } }$$, $$\frac { \sqrt { 3 } } { \sqrt { 7 } }$$, $$\frac { \sqrt { 3 } - \sqrt { 5 } } { \sqrt { 3 } }$$, $$\frac { \sqrt { 6 } - \sqrt { 2 } } { \sqrt { 2 } }$$, $$\frac { 3 b ^ { 2 } } { 2 \sqrt { 3 a b } }$$, $$\frac { 1 } { \sqrt [ 3 ] { 3 y ^ { 2 } } }$$, $$\frac { 9 x \sqrt[3] { 2 } } { \sqrt [ 3 ] { 9 x y ^ { 2 } } }$$, $$\frac { 5 y ^ { 2 } \sqrt [ 3 ] { x } } { \sqrt [ 3 ] { 5 x ^ { 2 } y } }$$, $$\frac { 3 a } { 2 \sqrt [ 3 ] { 3 a ^ { 2 } b ^ { 2 } } }$$, $$\frac { 25 n } { 3 \sqrt [ 3 ] { 25 m ^ { 2 } n } }$$, $$\frac { 3 } { \sqrt [ 5 ] { 27 x ^ { 2 } y } }$$, $$\frac { 2 } { \sqrt [ 5 ] { 16 x y ^ { 2 } } }$$, $$\frac { a b } { \sqrt [ 5 ] { 9 a ^ { 3 } b } }$$, $$\frac { a b c } { \sqrt [ 5 ] { a b ^ { 2 } c ^ { 3 } } }$$, $$\sqrt [ 5 ] { \frac { 3 x } { 8 y ^ { 2 } z } }$$, $$\sqrt [ 5 ] { \frac { 4 x y ^ { 2 } } { 9 x ^ { 3 } y z ^ { 4 } } }$$, $$\frac { 1 } { \sqrt { 5 } + \sqrt { 3 } }$$, $$\frac { 1 } { \sqrt { 7 } - \sqrt { 2 } }$$, $$\frac { \sqrt { 3 } } { \sqrt { 3 } + \sqrt { 6 } }$$, $$\frac { \sqrt { 5 } } { \sqrt { 5 } + \sqrt { 15 } }$$, $$\frac { - 2 \sqrt { 2 } } { 4 - 3 \sqrt { 2 } }$$, $$\frac { \sqrt { 3 } + \sqrt { 5 } } { \sqrt { 3 } - \sqrt { 5 } }$$, $$\frac { \sqrt { 10 } - \sqrt { 2 } } { \sqrt { 10 } + \sqrt { 2 } }$$, $$\frac { 2 \sqrt { 3 } - 3 \sqrt { 2 } } { 4 \sqrt { 3 } + \sqrt { 2 } }$$, $$\frac { 6 \sqrt { 5 } + 2 } { 2 \sqrt { 5 } - \sqrt { 2 } }$$, $$\frac { x - y } { \sqrt { x } + \sqrt { y } }$$, $$\frac { x - y } { \sqrt { x } - \sqrt { y } }$$, $$\frac { x + \sqrt { y } } { x - \sqrt { y } }$$, $$\frac { x - \sqrt { y } } { x + \sqrt { y } }$$, $$\frac { \sqrt { a } - \sqrt { b } } { \sqrt { a } + \sqrt { b } }$$, $$\frac { \sqrt { a b } + \sqrt { 2 } } { \sqrt { a b } - \sqrt { 2 } }$$, $$\frac { \sqrt { x } } { 5 - 2 \sqrt { x } }$$, $$\frac { \sqrt { x } + \sqrt { 2 y } } { \sqrt { 2 x } - \sqrt { y } }$$, $$\frac { \sqrt { 3 x } - \sqrt { y } } { \sqrt { x } + \sqrt { 3 y } }$$, $$\frac { \sqrt { 2 x + 1 } } { \sqrt { 2 x + 1 } - 1 }$$, $$\frac { \sqrt { x + 1 } } { 1 - \sqrt { x + 1 } }$$, $$\frac { \sqrt { x + 1 } + \sqrt { x - 1 } } { \sqrt { x + 1 } - \sqrt { x - 1 } }$$, $$\frac { \sqrt { 2 x + 3 } - \sqrt { 2 x - 3 } } { \sqrt { 2 x + 3 } + \sqrt { 2 x - 3 } }$$. Using the product rule for radicals and the fact that multiplication is commutative, we can multiply the coefficients and the radicands as follows. The radical in the denominator is equivalent to $$\sqrt [ 3 ] { 5 ^ { 2 } }$$. 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. A radical is a number or an expression under the root symbol. \begin{aligned} \sqrt [ 3 ] { 6 x ^ { 2 } y } \left( \sqrt [ 3 ] { 9 x ^ { 2 } y ^ { 2 } } - 5 \cdot \sqrt [ 3 ] { 4 x y } \right) & = \color{Cerulean}{\sqrt [ 3 ] { 6 x ^ { 2 } y }}\color{black}{\cdot} \sqrt [ 3 ] { 9 x ^ { 2 } y ^ { 2 } } - \color{Cerulean}{\sqrt [ 3 ] { 6 x ^ { 2 } y }}\color{black}{ \cdot} 5 \sqrt [ 3 ] { 4 x y } \\ & = \sqrt [ 3 ] { 54 x ^ { 4 } y ^ { 3 } } - 5 \sqrt [ 3 ] { 24 x ^ { 3 } y ^ { 2 } } \\ & = \sqrt [ 3 ] { 27 \cdot 2 \cdot x \cdot x ^ { 3 } \cdot y ^ { 3 } } - 5 \sqrt [ 3 ] { 8 \cdot 3 \cdot x ^ { 3 } \cdot y ^ { 2 } } \\ & = 3 x y \sqrt [ 3 ] { 2 x } - 10 x \sqrt [ 3 ] { 3 y ^ { 2 } } \\ & = 3 x y \sqrt [ 3 ] { 2 x } - 10 x \sqrt [ 3 ] { 3 y ^ { 2 } } \end{aligned}, $$3 x y \sqrt [ 3 ] { 2 x } - 10 x \sqrt [ 3 ] { 3 y ^ { 2 } }$$. We can use the property $$( \sqrt { a } + \sqrt { b } ) ( \sqrt { a } - \sqrt { b } ) = a - b$$ to expedite the process of multiplying the expressions in the denominator. $$\frac { \sqrt { 5 } - \sqrt { 3 } } { 2 }$$, 33. \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.} You multiply radical expressions that contain variables in the same manner. Apply the distributive property, and then simplify the result. Divide: \(\frac { \sqrt [ 3 ] { 96 } } { \sqrt [ 3 ] { 6 } }. Simplify each radical, if possible, before multiplying. $\begin{array}{l}12{{x}^{2}}\sqrt[4]{{{x}^{4}}\cdot {{y}^{4}}}\\12{{x}^{2}}\sqrt[4]{{{x}^{4}}}\cdot \sqrt[4]{{{y}^{4}}}\\12{{x}^{2}}\cdot \left| x \right|\cdot \left| y \right|\end{array}$. 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