Multiply: [tex]4 x \sqrt[3]{4 x^2}\left(2 \sqrt[3]{32 x^2}-x \sqrt[3]{2 x}\right)[/tex]
Answer (2)
To simplify the expression 4 x 3 4 x 2 ( 2 3 32 x 2 − x 3 2 x ) , we distribute and simplify to get 32 x 2 3 2 x − 8 x 3 . The final result is 32 x 2 3 2 x − 8 x 3 .
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Distribute the term 4 x 3 4 x 2 to both terms inside the parenthesis: 4 x 3 4 x 2 ∗ 2 3 32 x 2 − 4 x 3 4 x 2 ∗ x 3 2 x .
Simplify the first term: 4 x 3 4 x 2 ∗ 2 3 32 x 2 = 32 x 2 3 2 x .
Simplify the second term: 4 x 3 4 x 2 ∗ x 3 2 x = 8 x 3 .
Combine the simplified terms: 32 x 2 3 2 x − 8 x 3 . The final answer is 32 x 2 3 2 x − 8 x 3 .
Explanation
Understanding the Problem Let's analyze the given expression and plan our approach to simplify it. We have to multiply 4 x 3 4 x 2 with ( 2 3 32 x 2 − x 3 2 x ) . We will first distribute the term outside the parenthesis to each term inside the parenthesis and then simplify each of the resulting terms using properties of radicals and exponents.
Distributing the term First, distribute 4 x 3 4 x 2 to both terms inside the parentheses:
4 x 3 4 x 2 ⋅ 2 3 32 x 2 − 4 x 3 4 x 2 ⋅ x 3 2 x
Simplifying the first term Now, let's simplify the first term:
4 x 3 4 x 2 ⋅ 2 3 32 x 2 = 8 x 3 4 x 2 ⋅ 32 x 2 = 8 x 3 128 x 4
Since 128 = 2 7 , we can rewrite the expression as:
8 x 3 2 7 x 4 = 8 x 3 2 6 ⋅ 2 ⋅ x 3 ⋅ x = 8 x ⋅ 2 2 ⋅ x ⋅ 3 2 x = 32 x 2 3 2 x
Simplifying the second term Next, simplify the second term:
4 x 3 4 x 2 ⋅ x 3 2 x = 4 x 2 3 4 x 2 ⋅ 2 x = 4 x 2 3 8 x 3
Since 8 = 2 3 , we have:
4 x 2 3 2 3 x 3 = 4 x 2 ⋅ 2 x = 8 x 3
Combining the terms Now, combine the simplified terms:
32 x 2 3 2 x − 8 x 3
Final Answer So, the final simplified expression is:
32 x 2 3 2 x − 8 x 3
Examples
Imagine you are calculating the volume of a complex 3D shape that involves radicals and polynomial expressions. Simplifying such expressions, as we did here, allows you to efficiently compute the volume and optimize the design for various engineering applications. This type of algebraic manipulation is crucial in fields like mechanical engineering, where complex shapes and volumes need to be calculated accurately.
