Select the correct answer.
What is this expression in simplified form?
$\sqrt[3]{-128}$
A. $-4 \sqrt[3]{2}$
B. $-2 \sqrt[3]{4}$
C. $-4 \sqrt[3]{4}$
D. $-8 \sqrt[3]{2}$
Answer (2)
Express − 128 as − 1 ⋅ 2 7 .
Rewrite the expression as 3 − 1 ⋅ 2 7 .
Simplify the expression by taking out the perfect cubes: 3 − 1 ⋅ 2 6 ⋅ 2 = − 4 3 2 .
The simplified form of 3 − 128 is − 4 3 2 .
Explanation
Understanding the problem We are asked to simplify the cube root of -128, which is written as 3 − 128 .
Expressing -128 as a product of its prime factors First, we can express -128 as a product of its prime factors. We know that 128 = 2 7 , so − 128 = − 1 ⋅ 2 7 . Therefore, we can rewrite the expression as 3 − 1 ⋅ 2 7 .
Simplifying the expression Now, we simplify the expression by taking out the perfect cubes from under the cube root. We can rewrite 2 7 as 2 6 ⋅ 2 , which is ( 2 2 ) 3 ⋅ 2 = 4 3 ⋅ 2 . So, we have 3 − 128 = 3 − 1 ⋅ 2 7 = 3 − 1 ⋅ 2 6 ⋅ 2 = 3 − 1 ⋅ 4 3 ⋅ 2 Since 3 − 1 = − 1 and 3 4 3 = 4 , we can write 3 − 1 ⋅ 4 3 ⋅ 2 = − 1 ⋅ 4 ⋅ 3 2 = − 4 3 2 .
Final Answer Therefore, the simplified form of 3 − 128 is − 4 3 2 . Looking at the multiple choice options, we see that option A matches our simplified expression.
Examples
Cube roots are used in various fields like engineering and physics. For example, if you have a cube-shaped container with a volume of -128 cubic units, finding the cube root helps determine the length of one side of the container. In this case, the side length would be − 4 3 2 units. Understanding cube roots allows engineers to calculate dimensions and volumes accurately, ensuring designs are precise and functional.
The simplified form of 3 − 128 is − 4 3 2 , which corresponds to option A from the choices given. This is derived by expressing -128 as − 1 ⋅ 2 7 and simplifying the cube root. The final outcome of the simplification is − 4 3 2 .
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