Simplify. $\sqrt[5]{64 x^3 y^8}$
Answer (2)
Rewrite 64 as 2 6 and express the given expression as 5 2 6 x 3 y 8 .
Separate the terms inside the radical to identify multiples of 5 in the exponents: 5 2 5 ⋅ 2 x 3 y 5 ⋅ y 3 .
Simplify by taking out the terms with exponents that are multiples of 5: 2 y 5 2 x 3 y 3 .
The simplified expression is 2 y 5 2 x 3 y 3 .
Explanation
Understanding the Problem We are asked to simplify the expression 5 64 x 3 y 8 . Let's break down the terms inside the radical and see what we can extract.
Rewriting the Expression First, we can rewrite 64 as 2 6 . So the expression becomes 5 2 6 x 3 y 8 .
Finding Multiples of 5 Now, we want to find the largest multiples of 5 in the exponents of each term so we can take them out of the radical. We can rewrite the expression as 5 2 5 ⋅ 2 x 3 y 5 ⋅ y 3 .
Simplifying the Expression We can now simplify the expression by taking out the terms with exponents that are multiples of 5: 2 y 5 2 x 3 y 3 .
Comparing with Possible Answers Comparing this simplified expression with the possible answers, we see that it matches none of the provided options. However, let's re-examine our steps to ensure we haven't made a mistake. We have 5 64 x 3 y 8 = 5 2 6 x 3 y 8 = 5 2 5 ⋅ 2 ⋅ x 3 ⋅ y 5 ⋅ y 3 = 2 y 5 2 x 3 y 3 . It seems there was a typo in the original options. Let's assume the correct answer should be 2 y 5 2 x 3 y 3
Examples
Simplifying radicals is useful in various fields, such as engineering and physics, where complex equations need to be made more manageable. For example, when calculating the period of a pendulum or analyzing wave functions in quantum mechanics, simplifying radical expressions can lead to more understandable and solvable equations. This skill also helps in computer graphics when dealing with geometric transformations and scaling.
The expression 5 64 x 3 y 8 simplifies to 2 y 5 2 x 3 y 3 by rewriting the numbers as powers of their bases and extracting terms that are multiples of 5 from inside the radical. This process involves separating the terms and simplifying. Thus, the final answer is 2 y 5 2 x 3 y 3 .
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