What is the simplest form of $\sqrt[4]{324 x^6 y^8}$ ?
A. $3 x y^2 \sqrt[4]{2 x^2}$
B. $3 x y^2 \sqrt[4]{4 x^2}$
C. $6 x y \sqrt[4]{2 x}$
D. $6 x y \sqrt[4]{4 x^2}$
Answer (1)
Find the prime factorization of 324: 324 = 2 2 × 3 4 .
Rewrite the expression: 4 2 2 × 3 4 × x 6 × y 8 .
Simplify the expression: 3 x y 2 4 2 2 x 2 .
The simplest form is 3 x y 2 4 4 x 2 .
Explanation
Understanding the Problem We are given the expression 4 324 x 6 y 8 and we want to simplify it.
Prime Factorization First, let's find the prime factorization of 324. We know that 324 = 4 × 81 = 2 2 × 3 4 . So we can rewrite the expression as 4 2 2 × 3 4 × x 6 × y 8 .
Simplifying the Expression Now, we simplify the expression by taking out terms raised to the fourth power. We have 4 3 4 = 3 and 4 y 8 = y 2 . Also, we can rewrite x 6 as x 4 × x 2 , so 4 x 6 = 4 x 4 × x 2 = x 4 x 2 . Thus, the expression becomes 3 x y 2 4 2 2 x 2 .
Final Simplification Since 2 2 = 4 , we can write the simplified expression as 3 x y 2 4 4 x 2 .
Final Answer Therefore, the simplest form of 4 324 x 6 y 8 is 3 x y 2 4 4 x 2 .
Examples
Simplifying radicals is useful in various fields, such as engineering and physics, where complex calculations are often simplified to make them more manageable. For example, when calculating the area of a shape involving radicals, simplifying the radical expression can make the final result easier to understand and use. Imagine you are designing a square garden and the area is given by 4 324 x 6 y 8 . By simplifying this to 3 x y 2 4 4 x 2 , you can easily determine the side length of the garden for different values of x and y.
