[tex]\frac{125 x^8 y^4}{25 x^2 y^8}[/tex]
Answer (1)
Divide the coefficients: 25 125 = 5 .
Simplify the x terms: x 2 x 8 = x 8 − 2 = x 6 .
Simplify the y terms: y 8 y 4 = y 4 − 8 = y − 4 .
Combine the terms and rewrite with positive exponents: y 4 5 x 6 .
Explanation
Understanding the Problem We are given the expression 25 x 2 y 8 125 x 8 y 4 and our goal is to simplify it. We will simplify the coefficients and the variables separately, using the properties of exponents.
Simplifying Coefficients First, let's simplify the coefficients. We have 25 125 . Dividing 125 by 25, we get 5. So, 25 125 = 5 .
Simplifying x Terms Next, let's simplify the x terms. We have x 2 x 8 . Using the quotient rule for exponents, which states that x b x a = x a − b , we have x 2 x 8 = x 8 − 2 = x 6 .
Simplifying y Terms Now, let's simplify the y terms. We have y 8 y 4 . Using the same quotient rule for exponents, we have y 8 y 4 = y 4 − 8 = y − 4 .
Combining and Rewriting Finally, let's combine the simplified terms. We have 5 , x 6 , and y − 4 . Multiplying these together, we get 5 x 6 y − 4 . To write this with only positive exponents, we can rewrite y − 4 as y 4 1 . Therefore, the simplified expression is y 4 5 x 6 .
Examples
Simplifying algebraic expressions is a fundamental skill in mathematics and has numerous applications in real-world scenarios. For example, engineers use simplified expressions to model physical systems, such as electrical circuits or mechanical structures. By simplifying the expressions, they can more easily analyze the behavior of the system and design it to meet specific requirements. Similarly, in computer graphics, simplified expressions are used to optimize rendering algorithms, allowing for faster and more efficient generation of images and animations. This ensures smoother and more responsive user experiences in applications like video games and virtual reality.
