An expression is given.
[tex]\left(-6 x^2 y^{-3}\right)\left(-2 x^{-4} y^{-3}\right)[/tex]
It has been simplified and written using the powers with positive exponents:
[tex]\frac{12}{x^m y^n}[/tex]
Determine the value of [tex]m[/tex] and [tex]n[/tex]. Record as one value (side by side): [tex]m n[/tex].
For example, if [tex]m = 1[/tex] and [tex]n = 3[/tex], state the answer as [tex]1 3[/tex].
Answer (1)
Multiply the coefficients: − 6 × − 2 = 12 .
Multiply the x terms: x 2 × x − 4 = x − 2 .
Multiply the y terms: y − 3 × y − 3 = y − 6 .
Rewrite the expression with positive exponents: 12 x − 2 y − 6 = x 2 y 6 12 , so m = 2 and n = 6 . The final answer is 26 .
Explanation
Understanding the Problem We are given the expression ( − 6 x 2 y − 3 ) ( − 2 x − 4 y − 3 ) and we want to simplify it to the form x m y n 12 , where m and n are positive integers. Our goal is to find the values of m and n and then write them side by side as a single number.
Multiplying the Coefficients First, we multiply the coefficients: − 6 × − 2 = 12 .
Multiplying the x Terms Next, we multiply the x terms: x 2 × x − 4 = x 2 + ( − 4 ) = x − 2 .
Multiplying the y Terms Then, we multiply the y terms: y − 3 × y − 3 = y − 3 + ( − 3 ) = y − 6 .
Combining the Results Now, we combine the results: 12 x − 2 y − 6 .
Rewriting with Positive Exponents We rewrite the expression with positive exponents: 12 x − 2 y − 6 = x 2 y 6 12 .
Determining m and n Comparing the simplified expression x 2 y 6 12 with the given form x m y n 12 , we can determine the values of m and n . We have m = 2 and n = 6 .
Recording the Final Answer Finally, we record the values of m and n as a single number mn = 26 .
Examples
Understanding how to simplify expressions with exponents is crucial in many fields, such as physics and engineering. For example, when calculating the force between two charged particles, you need to manipulate expressions with negative exponents. Simplifying these expressions correctly ensures accurate calculations and predictions in real-world applications.
