Express the answer in simplest radical form.
[tex]$\sqrt[3]{189 x^7 y^8}$[/tex]
Answer (2)
Find the prime factorization of 189: 189 = 3 3 × 7 .
Rewrite the expression: 3 189 x 7 y 8 = 3 3 3 ⋅ 7 ⋅ x 6 ⋅ x ⋅ y 6 ⋅ y 2 .
Simplify by taking out perfect cubes: 3 x 2 y 2 3 7 x y 2 .
The simplified expression is 3 x 2 y 2 3 7 x y 2 .
Explanation
Understanding the Problem We are asked to simplify the cube root of 189 x 7 y 8 and express the answer in simplest radical form.
Prime Factorization First, we find the prime factorization of 189. The prime factorization of 189 is 3 × 3 × 3 × 7 = 3 3 × 7 .
Rewriting the Expression Now we rewrite the expression as 3 3 3 ⋅ 7 ⋅ x 7 ⋅ y 8 . We can further break down the variables as 3 3 3 ⋅ 7 ⋅ x 6 ⋅ x ⋅ y 6 ⋅ y 2 .
Simplifying the Cube Root Next, we simplify the cube root by taking out perfect cubes. We have 3 3 3 ⋅ 7 ⋅ x 6 ⋅ x ⋅ y 6 ⋅ y 2 = 3 3 3 ⋅ 3 x 6 ⋅ 3 y 6 ⋅ 3 7 x y 2 = 3 ⋅ x 2 ⋅ y 2 ⋅ 3 7 x y 2 .
Final Answer Therefore, the simplified expression is 3 x 2 y 2 3 7 x y 2 .
Examples
Simplifying radicals is useful in various fields, such as engineering and physics, when dealing with measurements or calculations involving irrational numbers. For instance, when calculating the volume of a cube with side length 3 189 x 7 y 8 , expressing the side length in simplest radical form allows for easier manipulation and understanding of the volume's components. This ensures accurate and efficient problem-solving in practical applications.
To simplify 3 189 x 7 y 8 , we first find the prime factorization of 189 and rewrite the expression. By extracting the perfect cubes, we find that the answer in simplest radical form is 3 x 2 y 2 3 7 x y 2 .
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