What is the following quotient? [tex]\frac{\sqrt[3]{60}}{\sqrt[3]{20}}[/tex]
Answer (2)
The quotient 3 20 3 60 simplifies to 3 3 by applying the property of roots. This involves first rewriting the expression using the quotient rule for roots and then simplifying the fraction. Therefore, the final answer is 3 3 .
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Use the property n b n a = n b a to rewrite the expression as 3 20 60 .
Simplify the fraction inside the cube root: 20 60 = 3 .
The expression simplifies to 3 3 .
The final answer is 3 3 .
Explanation
Understanding the problem We are asked to find the quotient of two cube roots: 3 20 3 60 .
Applying the quotient rule for roots To simplify this expression, we can use the property that the quotient of two roots with the same index is equal to the root of the quotient. In other words, n b n a = n b a . Applying this property, we get: 3 20 3 60 = 3 20 60
Simplifying the fraction Now, we simplify the fraction inside the cube root: 20 60 = 3
Finding the final result Substituting this back into our expression, we have: 3 20 60 = 3 3
Conclusion Therefore, the quotient 3 20 3 60 simplifies to 3 3 .
Examples
Cube roots are useful in various fields, such as engineering and physics, when dealing with volumes. For example, if you have a cube with a volume of 60 cubic units and you want to compare its side length to a cube with a volume of 20 cubic units, you would calculate the quotient of their cube roots, as we did in this problem. This allows you to directly compare the relative sizes of the cubes' sides. Understanding how to simplify and compare cube roots can help in optimizing designs and understanding physical properties in these fields.
