Simplify the following expression: $ne^{mx+nx} - me^{mx-nx} - ne^{mx-nx} - me^{mx+nx}$
Understand the Problem
The question presents an expression involving exponential terms with 'e' raised to linear functions of 'x', combined with constants 'm' and 'n'. The task likely involves simplifying or manipulating this expression through algebraic techniques such as factoring, combining like terms, or applying exponential identities.
Answer
$(n-m)e^{mx+nx} - (m+n)e^{mx-nx}$
Answer for screen readers
$(n-m)e^{mx+nx} - (m+n)e^{mx-nx}$
Steps to Solve
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Rewrite the expression The given expression is: $ne^{mx+nx} - me^{mx-nx} - ne^{mx-nx} - me^{mx+nx}$
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Group like terms Group the terms with the same exponential part: $(ne^{mx+nx} - me^{mx+nx}) + (-me^{mx-nx} - ne^{mx-nx})$
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Factor out the exponential terms Factor out the exponential terms from each group: $(n-m)e^{mx+nx} + (-m-n)e^{mx-nx}$
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Simplify the second term Rewrite the second term by factoring out a negative sign: $(n-m)e^{mx+nx} - (m+n)e^{mx-nx}$
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Final Answer The simplified expression is: $(n-m)e^{mx+nx} - (m+n)e^{mx-nx}$
$(n-m)e^{mx+nx} - (m+n)e^{mx-nx}$
More Information
This simplification demonstrates combining like terms and factoring in expressions with exponential functions, which is a common technique in calculus and differential equations
Tips
A common mistake is to incorrectly group the terms or to make errors when factoring out the exponential terms. For example, a mistake might be forgetting to factor out the negative sign in the second part of the expression.
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