Write in simplified radical form with at most one radical.
[tex]$\sqrt[4]{32 x y^7} \sqrt{2 x^5 y}$[/tex]
Assume that the variables represent positive real numbers.
Answer (2)
Rewrite the radicals using fractional exponents.
Combine like terms by adding exponents.
Rewrite the expression with a single radical.
Simplify the radical by extracting perfect fourth powers: 2 x 2 y 2 4 8 x 3 y .
Explanation
Understanding the Problem We are given the expression 4 32 x y 7 2 x 5 y , where x and y are positive real numbers. Our goal is to simplify this expression into a single radical in simplified form.
Rewriting with Fractional Exponents First, let's rewrite the radicals using fractional exponents. Recall that n a = a 1/ n . Thus, we have 4 32 x y 7 = ( 32 x y 7 ) 1/4 2 x 5 y = ( 2 x 5 y ) 1/2 So the expression becomes ( 32 x y 7 ) 1/4 ( 2 x 5 y ) 1/2 .
Distributing the Exponents Now, let's rewrite 32 as 2 5 . The expression is now ( 2 5 x y 7 ) 1/4 ( 2 x 5 y ) 1/2 . Distribute the exponents to each term inside the parentheses: 2 5/4 x 1/4 y 7/4 ⋅ 2 1/2 x 5/2 y 1/2 .
Combining Like Terms Next, we combine like terms by adding the exponents. For the constant 2, we have 2 5/4 ⋅ 2 1/2 = 2 5/4 + 1/2 = 2 5/4 + 2/4 = 2 7/4 . For x , we have x 1/4 ⋅ x 5/2 = x 1/4 + 5/2 = x 1/4 + 10/4 = x 11/4 . For y , we have y 7/4 ⋅ y 1/2 = y 7/4 + 1/2 = y 7/4 + 2/4 = y 9/4 .
So the expression is now 2 7/4 x 11/4 y 9/4 .
Rewriting with a Single Radical Now, let's rewrite the expression with a single radical. We have 2 7/4 x 11/4 y 9/4 = ( 2 7 x 11 y 9 ) 1/4 = 4 2 7 x 11 y 9 .
Simplifying the Radical Finally, we simplify the radical by extracting perfect fourth powers. We have 4 2 7 x 11 y 9 = 4 2 4 ⋅ 2 3 ⋅ x 8 ⋅ x 3 ⋅ y 8 ⋅ y = 4 2 4 x 8 y 8 ⋅ 2 3 x 3 y = 2 x 2 y 2 4 2 3 x 3 y = 2 x 2 y 2 4 8 x 3 y .
Final Answer Therefore, the simplified expression is 2 x 2 y 2 4 8 x 3 y .
Examples
Imagine you are calculating the dimensions of a rectangular prism where the volume involves radicals. Simplifying these radicals, as we did here, makes it easier to understand and work with the actual measurements. For instance, if the volume is given by an expression like 4 32 x y 7 2 x 5 y , simplifying it to 2 x 2 y 2 4 8 x 3 y allows for easier calculation of the side lengths or other properties of the prism. This type of simplification is also useful in physics when dealing with quantities involving roots, such as in wave mechanics or electromagnetism.
The expression 4 32 x y 7 2 x 5 y simplifies to 2 x 2 y 2 4 8 x 3 y by rewriting radicals as fractional exponents, combining like terms, and extracting perfect fourth powers.
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