Combine and simplify the following radical expression: $(2 \sqrt[3]{12})(3 \sqrt[3]{2})$
Answer (2)
To combine and simplify the expression ( 2 3 12 ) ( 3 3 2 ) , we multiply the coefficients and the radicands, leading to a final result of 12 3 3 .
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Multiply the coefficients: 2 ⋅ 3 = 6 .
Multiply the radicands: 12 ⋅ 2 = 24 , resulting in 6 3 24 .
Factor the radicand: 24 = 2 3 ⋅ 3 , so we have 6 3 2 3 ⋅ 3 .
Simplify: 6 ⋅ 2 3 3 = 12 3 3 . The final answer is 12 3 3 .
Explanation
Understanding the problem We are given the expression ( 2 3 12 ) ( 3 3 2 ) . Our goal is to simplify this expression by combining the terms under the radicals and reducing it to its simplest form.
Multiplying Coefficients First, we multiply the coefficients outside the radicals: 2 ⋅ 3 = 6 .
Multiplying Radicands Next, we multiply the radicands inside the cube roots: 12 ⋅ 2 = 24 . So now we have 6 3 24 .
Prime Factorization Now, we find the prime factorization of 24. We can write 24 = 2 3 ⋅ 3 .
Rewriting the Expression We rewrite the expression as 6 3 2 3 ⋅ 3 .
Simplifying the Cube Root Now we simplify the cube root. Since 3 2 3 = 2 , we have 3 2 3 ⋅ 3 = 2 3 3 .
Final Simplification Finally, we multiply the coefficient outside the radical: 6 ⋅ 2 3 3 = 12 3 3 . Therefore, the simplified expression is 12 3 3 .
Examples
Radical expressions are useful in various fields, such as engineering and physics, when dealing with quantities that involve roots, like calculating the period of a pendulum or determining the dimensions of a structure. For instance, if you are designing a cubic container with a volume of V = x 3 , and you need to find the length of one side, you would use a cube root, x = 3 V . Simplifying radical expressions helps in obtaining precise and manageable values for these calculations.
