Look at this expression.
[tex]24 x^4 y[/tex]
[tex]3 x^{-4} y^{-1}[/tex]
Which of the following is the simplest form of the expression above?
A. 8
B. [tex]8 x^{-16} y^{-1}[/tex]
C. [tex]8 x^{-1} y^{-1}[/tex]
D. [tex]8 x^8 y^2[/tex]
Answer (2)
Divide the coefficients: 3 24 = 8 .
Divide the x terms: x − 4 x 4 = x 4 − ( − 4 ) = x 8 .
Divide the y terms: y − 1 y = y 1 − ( − 1 ) = y 2 .
Combine the results: The simplest form of the expression is 8 x 8 y 2 .
Explanation
Understanding the Problem We are given two expressions: 24 x 4 y and 3 x − 4 y − 1 . The problem asks us to simplify the expression obtained by dividing the first expression by the second one.
Plan for Simplification To simplify the expression 3 x − 4 y − 1 24 x 4 y , we will divide the coefficients and use the exponent rules to simplify the variables.
Dividing the Coefficients First, divide the coefficients: 3 24 = 8
Dividing the x Terms Next, divide the x terms. Recall that when dividing terms with exponents, we subtract the exponents: x − 4 x 4 = x 4 − ( − 4 ) = x 4 + 4 = x 8
Dividing the y Terms Now, divide the y terms: y − 1 y = y 1 − ( − 1 ) = y 1 + 1 = y 2
Combining the Results Finally, combine the results to get the simplified expression: 8 x 8 y 2
Final Answer Therefore, the simplest form of the expression is 8 x 8 y 2 , which corresponds to option D.
Examples
Understanding how to simplify expressions with exponents is crucial in many areas of science and engineering. For example, in physics, when dealing with quantities like force or energy that depend on distance, you might encounter expressions involving exponents. Simplifying these expressions allows you to easily analyze how these quantities change with distance. In computer graphics, transformations like scaling and rotations often involve matrix operations that can be simplified using exponent rules, making calculations more efficient.
The simplest form of the expression 3 x − 4 y − 1 24 x 4 y is 8 x 8 y 2 , which corresponds to option D. We achieve this by dividing the coefficients, the x terms, and the y terms, and then combining the results. Thus, the answer is option D: 8 x 8 y 2 .
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