What is the following quotient?
[tex]$\frac{\sqrt{96}}{\sqrt{8}}$[/tex]
Answer (2)
The quotient 8 96 simplifies to 2 3 . This is done by using the property of square roots to combine them, simplifying the fraction inside the root, and breaking down the square root of 12. The final answer is 2 3 .
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Use the property b a = b a to rewrite the expression as 8 96 .
Simplify the fraction inside the square root: 8 96 = 12 .
Simplify the square root: 12 = 4 ⋅ 3 = 4 ⋅ 3 = 2 3 .
The final answer is 2 3 .
Explanation
Understanding the Problem We are asked to find the quotient of two square roots: 8 96 . We need to simplify this expression and choose the correct answer from the given options.
Applying the Square Root Property We can use the property of square roots that states b a = b a . Applying this property, we get: 8 96 = 8 96
Simplifying the Fraction Now, we simplify the fraction inside the square root: 8 96 = 12 So, we have: 8 96 = 12
Simplifying the Square Root Next, we simplify the square root of 12. We can write 12 as a product of its prime factors: 12 = 4 × 3 . Therefore, 12 = 4 × 3 = 4 × 3 = 2 3
Final Answer Thus, the quotient 8 96 simplifies to 2 3 .
Examples
Understanding how to simplify quotients of square roots is useful in various fields, such as physics and engineering, when dealing with calculations involving distances, areas, or volumes. For example, if you are calculating the ratio of two lengths that are expressed as square roots, simplifying the quotient can make the calculations easier and more intuitive. Imagine you are comparing the lengths of two sides of a rectangular garden, where one side has a length of 96 meters and the other has a length of 8 meters. By simplifying the quotient 8 96 to 2 3 , you can quickly understand the relationship between the lengths of the two sides.
