Simplify the following expression completely: [tex]$\frac{x^6 y^4}{x y^2}$[/tex]
Answer (2)
The simplified expression for x y 2 x 6 y 4 is x 5 y 2 . This was achieved by applying the quotient rule for exponents to both the x and y terms. The final result combines the simplified x and y terms together.
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Rewrite the expression as a product of quotients: x y 2 x 6 y 4 = x x 6 ⋅ y 2 y 4 .
Apply the quotient rule to simplify the x terms: x x 6 = x 6 − 1 = x 5 .
Apply the quotient rule to simplify the y terms: y 2 y 4 = y 4 − 2 = y 2 .
Combine the simplified terms to get the final answer: x 5 y 2 .
Explanation
Understanding the Problem We are given the expression x y 2 x 6 y 4 to simplify. Our goal is to simplify it completely. We will use the quotient rule for exponents: a n a m = a m − n .
Rewriting the Expression We can rewrite the expression as a product of quotients: x y 2 x 6 y 4 = x x 6 ⋅ y 2 y 4
Simplifying x terms Now, we apply the quotient rule to the x terms: x x 6 = x 6 − 1 = x 5
Simplifying y terms Next, we apply the quotient rule to the y terms: y 2 y 4 = y 4 − 2 = y 2
Combining the terms Finally, we combine the simplified x and y terms: x y 2 x 6 y 4 = x 5 y 2
Final Answer Therefore, the simplified expression is x 5 y 2 .
Examples
Imagine you're organizing a rectangular garden. You know the total area is represented by x 6 y 4 square feet, and you want to divide it into smaller, identical plots each with an area of x y 2 square feet. Simplifying the expression x y 2 x 6 y 4 helps you determine how many of these smaller plots you can create. In this case, you can create x 5 y 2 smaller plots. This kind of simplification is useful in resource allocation, spatial planning, and understanding proportional relationships.
