What is the product of [tex]$\left(5 m^{-2}\right)\left(2 m^{-3}\right)$[/tex]?
Answer (2)
The product of ( 5 m − 2 ) ( 2 m − 3 ) equals m 5 10 . This is achieved by multiplying the coefficients and adding the exponents of the variable. The final result shows how negative exponents translate to a reciprocal form.
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Multiply the coefficients: 5 × 2 = 10 .
Multiply the variables by adding the exponents: m − 2 × m − 3 = m − 5 .
Combine the results: 10 m − 5 .
Rewrite the expression with a positive exponent: m 5 10 . The final answer is m 5 10 .
Explanation
Understanding the problem We are asked to find the product of two expressions: ( 5 m − 2 ) and ( 2 m − 3 ) . To do this, we will multiply the coefficients and add the exponents of the variable m .
Multiplying the coefficients First, let's multiply the coefficients: 5 × 2 = 10 .
Multiplying the variables Next, let's multiply the variables. When multiplying variables with exponents, we add the exponents: m − 2 × m − 3 = m − 2 + ( − 3 ) = m − 5 .
Combining the results Now, let's combine the results: 10 m − 5 . Since m − 5 = m 5 1 , we can rewrite the expression as m 5 10 .
Final Answer Therefore, the product of ( 5 m − 2 ) ( 2 m − 3 ) is m 5 10 .
Examples
Understanding exponents is crucial in many fields, such as calculating the size of computer files or understanding exponential growth in biology. For example, when measuring computer memory, we often use kilobytes (KB), megabytes (MB), and gigabytes (GB), which are powers of 2 (e.g., 1 KB = 2 10 bytes). Similarly, in finance, compound interest can be calculated using exponents, where the initial investment grows exponentially over time. This problem reinforces the rules of exponents, which are fundamental in these applications.
