What is the following quotient?
$\frac{\sqrt[3]{60}}{\sqrt[3]{20}}$
Answer (2)
Use the property of radicals to rewrite the quotient as a single cube root: 3 20 3 60 = 3 20 60 .
Simplify the fraction inside the cube root: 20 60 = 3 .
Evaluate the cube root: 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 . To simplify this expression, we can use the property that n b n a = n b a .
Rewriting the Expression Using the property of radicals, we can rewrite the expression as: 3 20 3 60 = 3 20 60 .
Simplifying the Fraction Now, we simplify the fraction inside the cube root: 20 60 = 3 So, the expression becomes: 3 3
Finding the Quotient Therefore, the quotient is 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 create a smaller cube with a volume of 20 cubic units, the ratio of their side lengths would be 3 20 3 60 = 3 3 . This tells you how much smaller the side length of the second cube is compared to the first.
The quotient of 3 20 3 60 simplifies to 3 3 using properties of radicals. This is done by rewriting the quotient as a single cube root and simplifying the fraction inside. Therefore, the final answer is 3 3 .
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