Simplify The Following Expression: M R + ( M − I ) 5 N = ( M − 2 ) 10 N , Where N ≠ 0 \frac{m}{R} + \frac{(m-i)}{5n} = \frac{(m-2)}{10n}, \text{ Where } N \neq 0 R M ​ + 5 N ( M − I ) ​ = 10 N ( M − 2 ) ​ , Where N  = 0

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Introduction

In this article, we will simplify the given expression mR+(m−i)5n=(m−2)10n\frac{m}{R} + \frac{(m-i)}{5n} = \frac{(m-2)}{10n}, where n≠0n \neq 0. This expression involves fractions and variables, and our goal is to simplify it to its simplest form.

Understanding the Expression

The given expression is a linear equation involving fractions. It consists of three terms: mR\frac{m}{R}, (m−i)5n\frac{(m-i)}{5n}, and (m−2)10n\frac{(m-2)}{10n}. The first term is a fraction with mm as the numerator and RR as the denominator. The second term is also a fraction, but with (m−i)(m-i) as the numerator and 5n5n as the denominator. The third term is a fraction with (m−2)(m-2) as the numerator and 10n10n as the denominator.

Simplifying the Expression

To simplify the expression, we need to get rid of the fractions. We can do this by multiplying both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the LCM of RR, 5n5n, and 10n10n is 10nR10nR.

from sympy import symbols, Eq, solve

m, R, i, n = symbols('m R i n')



eq = Eq(m/R + (m-i)/(5n), (m-2)/(10n))



eq = Eq(10nR*(m/R + (m-i)/(5n)), 10nR((m-2)/(10*n)))

Expanding and Simplifying

After multiplying both sides by 10nR10nR, we can expand and simplify the equation.

# Expand and simplify the equation
eq = Eq(10*n*R*m/R + 10*n*R*(m-i)/(5*n), 10*n*R*(m-2)/(10*n))

Canceling Out Common Factors

We can cancel out common factors in the equation to simplify it further.

# Cancel out common factors
eq = Eq(2*n*R*m + 2*n*R*(m-i), n*R*(m-2))

Combining Like Terms

We can combine like terms in the equation to simplify it further.

# Combine like terms
eq = Eq(2*n*R*m + 2*n*R*m - 2*n*R*i, n*R*m - 2*n*R)

Simplifying Further

We can simplify the equation further by combining like terms.

# Simplify further
eq = Eq(4*n*R*m - 2*n*R*i, n*R*m - 2*n*R)

Isolating the Variable

We can isolate the variable mm by moving all the terms involving mm to one side of the equation.

# Isolate the variable m
eq = Eq(4*n*R*m - n*R*m, 2*n*R - 2*n*R*i)

Simplifying the Equation

We can simplify the equation by combining like terms.

# Simplify the equation
eq = Eq(3*n*R*m, 2*n*R*(1 - i))

Solving for the Variable

We can solve for the variable mm by dividing both sides of the equation by 3nR3nR.

# Solve for the variable m
eq = Eq(m, (2*n*R*(1 - i))/(3*n*R))

Simplifying the Solution

We can simplify the solution by canceling out common factors.

# Simplify the solution
eq = Eq(m, (2*(1 - i))/3)

Conclusion

In this article, we simplified the given expression mR+(m−i)5n=(m−2)10n\frac{m}{R} + \frac{(m-i)}{5n} = \frac{(m-2)}{10n}, where n≠0n \neq 0. We used algebraic manipulations to simplify the expression and isolate the variable mm. The final solution is m=2(1−i)3m = \frac{2(1 - i)}{3}.

Final Answer

The final answer is 2(1−i)3\boxed{\frac{2(1 - i)}{3}}.

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Introduction

In our previous article, we simplified the given expression mR+(m−i)5n=(m−2)10n\frac{m}{R} + \frac{(m-i)}{5n} = \frac{(m-2)}{10n}, where n≠0n \neq 0. In this article, we will answer some frequently asked questions (FAQs) related to the simplification of the expression.

Q&A

Q: What is the least common multiple (LCM) of the denominators in the given expression?

A: The LCM of the denominators RR, 5n5n, and 10n10n is 10nR10nR.

Q: Why do we need to multiply both sides of the equation by the LCM of the denominators?

A: We need to multiply both sides of the equation by the LCM of the denominators to get rid of the fractions.

Q: How do we simplify the expression after multiplying both sides by the LCM of the denominators?

A: We can simplify the expression by expanding and canceling out common factors.

Q: What is the final solution for the variable mm?

A: The final solution for the variable mm is m=2(1−i)3m = \frac{2(1 - i)}{3}.

Q: What is the significance of the variable ii in the final solution?

A: The variable ii represents the imaginary unit, which is defined as the square root of −1-1. In the final solution, ii is used to simplify the expression.

Q: Can we simplify the expression further?

A: Yes, we can simplify the expression further by combining like terms and canceling out common factors.

Q: What is the importance of simplifying the expression?

A: Simplifying the expression is important because it helps us to understand the underlying structure of the equation and to identify the relationships between the variables.

Common Mistakes

Mistake 1: Not multiplying both sides of the equation by the LCM of the denominators

A: Not multiplying both sides of the equation by the LCM of the denominators can lead to incorrect simplification of the expression.

Mistake 2: Not canceling out common factors

A: Not canceling out common factors can lead to unnecessary complexity in the expression.

Mistake 3: Not combining like terms

A: Not combining like terms can lead to unnecessary complexity in the expression.

Conclusion

In this article, we answered some frequently asked questions (FAQs) related to the simplification of the expression mR+(m−i)5n=(m−2)10n\frac{m}{R} + \frac{(m-i)}{5n} = \frac{(m-2)}{10n}, where n≠0n \neq 0. We also discussed some common mistakes that can occur during the simplification process.

Final Answer

The final answer is 2(1−i)3\boxed{\frac{2(1 - i)}{3}}.