What is the following product?
[tex]$\sqrt[3]{16 x^7} \cdot \sqrt[3]{12 x^9}$[/tex]
A. [tex]$x^2(\sqrt[3]{28 x^2})$[/tex]
B. [tex]$x^5(\sqrt[3]{28 x})$[/tex]
C. [tex]$4 x^2(\sqrt[3]{3 x^2})$[/tex]
D. [tex]$4 x^5(\sqrt[3]{3 x})$[/tex]
Answer (2)
Combine the cube roots: 3 16 x 7 ⋅ 3 12 x 9 = 3 ( 16 x 7 ) ( 12 x 9 ) .
Simplify the expression inside the cube root: ( 16 x 7 ) ( 12 x 9 ) = 192 x 16 .
Simplify the cube root: 3 192 x 16 = 3 2 6 ⋅ 3 ⋅ x 15 ⋅ x = 4 x 5 3 3 x .
The simplified expression is: 4 x 5 ( 3 3 x ) .
Explanation
Understanding the Problem We are given the expression 3 16 x 7 ⋅ 3 12 x 9 and five possible simplifications. We want to determine which of the five options is the correct simplification of the given expression.
Combining Cube Roots First, we combine the two cube roots into a single cube root: 3 16 x 7 ⋅ 3 12 x 9 = 3 ( 16 x 7 ) ( 12 x 9 ) .
Simplifying the Expression Next, we simplify the expression inside the cube root: ( 16 x 7 ) ( 12 x 9 ) = ( 16 ⋅ 12 ) ( x 7 ⋅ x 9 ) = 192 x 16 .
Rewriting the Expression Now we rewrite the expression as 3 192 x 16 .
Prime Factorization We factor 192 into its prime factorization: 192 = 2 6 ⋅ 3 = 64 ⋅ 3 .
Rewriting the Exponent We rewrite x 16 as x 15 ⋅ x = ( x 5 ) 3 ⋅ x .
Rewriting the Cube Root Now we rewrite the cube root as 3 2 6 ⋅ 3 ⋅ x 15 ⋅ x = 3 ( 2 2 ) 3 ⋅ 3 ⋅ ( x 5 ) 3 ⋅ x = 3 4 3 ⋅ ( x 5 ) 3 ⋅ 3 x .
Simplifying the Cube Root Finally, we simplify the cube root: 3 4 3 ⋅ ( x 5 ) 3 ⋅ 3 x = 4 x 5 3 3 x .
Identifying the Correct Option Comparing the simplified expression 4 x 5 3 3 x with the given options, we identify the correct one.
Examples
Cube roots are used in various fields such as engineering and physics to calculate volumes and dimensions. For example, if you have a cube-shaped container and you know its volume, you can use a cube root to find the length of one of its sides. This is also applicable in acoustics when dealing with sound intensity and pressure calculations, where cube roots help in determining the actual physical quantities.
The simplified expression of 3 16 x 7 ⋅ 3 12 x 9 is 4 x 5 3 3 x . Therefore, the correct answer is option D.
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