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(3m)/(4m-n)×((4m-n)^(2))/(6mn)

$\frac{3 m}{4 m-n} \times \frac{(4 m-n)^{2}}{6 m n}$

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Divide polynomials using long division

Full solution

Q.

\frac{3 m}{4 m-n} \times \frac{(4 m-n)^{2}}{6 m n}

Write Original Expression: Write down the original expression. $\newline$ We have the expression $(\frac{3m}{4m-n})\times\left(\frac{(4m-n)^{2}}{6mn}\right)$ .
Factor Out Common Terms: Factor out the common terms in the numerator and the denominator. $\newline$ Notice that $(4m-n)$ in the numerator and the denominator can be simplified. $\newline$ $\frac{3m}{4m-n}\times\frac{(4m-n)^{2}}{6mn}$ = $\frac{3m \cdot (4m-n) \cdot (4m-n)}{(4m-n) \cdot 6mn}$
Cancel Common Term: Cancel out the common $(4m-n)$ term from the numerator and the denominator. $\newline$ $(3m \times (4m-n) \times (4m-n)) / ((4m-n) \times 6mn) = (3m \times (4m-n)) / (6mn)$
Simplify Expression: Simplify the expression by canceling out common factors. $\newline$ $(3m \times (4m-n)) / (6mn) = (\frac{3}{6}) \times (\frac{m}{n}) \times (4m-n)$
Reduce Fraction: Reduce the fraction $(\frac{3}{6})$ to its simplest form. $(\frac{3}{6}) \times (\frac{m}{n}) \times (4m-n) = (\frac{1}{2}) \times (\frac{m}{n}) \times (4m-n)$
Multiply Remaining Terms: Multiply the remaining terms. $\newline$ $(\frac{1}{2}) \times (\frac{m}{n}) \times (4m-n) = \frac{2m^2 - mn}{2n}$
Check Further Simplification: Check for any further simplification. $\newline$ The expression $(2m^2 - mn) / (2n)$ is already in its simplest form.

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