What is the simplest form of [tex]$\sqrt[4]{81 x^8 y^5}$[/tex]?
Answer (2)
The simplest form of 4 81 x 8 y 5 is 3 x 2 y 4 y . This is achieved by breaking it down into its components, rewriting them with fractional exponents, and applying simplification rules. The process illustrates how to handle constants and variable exponents effectively.
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Rewrite the expression using fractional exponents: ( 81 x 8 y 5 ) 4 1 .
Simplify the constant: ( 3 4 x 8 y 5 ) 4 1 .
Simplify the variables: 3 x 2 y 4 y .
The simplest form is 3 x 2 y 4 y .
Explanation
Problem Analysis We are given the expression 4 81 x 8 y 5 and we want to simplify it.
Rewrite the expression First, we can rewrite the expression as ( 81 x 8 y 5 ) 4 1 .
Rewrite 81 We know that 81 = 3 4 , so we can rewrite the expression as ( 3 4 x 8 y 5 ) 4 1 .
Rewrite the variables We can also rewrite x 8 as ( x 2 ) 4 . And y 5 can be written as y 4 ⋅ y . So the expression becomes ( 3 4 ( x 2 ) 4 y 4 y ) 4 1 .
Apply the exponent Now, we apply the exponent 4 1 to each term inside the parenthesis: 3 4 ⋅ 4 1 ( x 2 ) 4 ⋅ 4 1 y 4 ⋅ 4 1 y 4 1 .
Simplify the exponents Simplifying the exponents, we get 3 1 x 2 y 1 y 4 1 , which is 3 x 2 y y 4 1 .
Rewrite the fractional exponent Finally, we rewrite y 4 1 as 4 y . Therefore, the simplified form is 3 x 2 y 4 y .
Final Answer Thus, the simplest form of 4 81 x 8 y 5 is 3 x 2 y 4 y .
Examples
Imagine you are calculating the dimensions of a storage container. If the volume of the container is expressed as 4 81 x 8 y 5 , simplifying it to 3 x 2 y 4 y helps you understand the relationship between the variables and optimize the container's design. This simplification allows for easier calculations and a better understanding of how changes in x and y affect the overall volume, which is crucial for efficient storage solutions.
