This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To simplify the given expression, we will factorize each polynomial and then perform the division. The expression is: (m^2 - n^2)/(m^2 - 2mn + n^2) ÷ (m^2 + mn)/(n^2 - mn) Step 1: Factorize each part of the fractions. • Numerator of the first fraction: m^2 - n^2 This is a difference of squares, which factors as (m-n)(m+n). • Denominator of the first fraction: m^2 - 2mn + n^2 This is a perfect square trinomial, which factors as (m-n)^2. • Numerator of the second fraction: m^2 + mn Factor out the common term m: m(m+n). • Denominator of the second fraction: n^2 - mn Factor out the common term n: n(n-m). We can rewrite (n-m) as -(m-n), so n(n-m) = -n(m-n). Step 2: Substitute the factored forms back into the expression. ((m-n)(m+n))/((m-n)^2) ÷ (m(m+n))/(-n(m-n)) Step 3: Change the division to multiplication by taking the reciprocal of the second fraction. ((m-n)(m+n))/((m-n)^2) × (-n(m-n))/(m(m+n)) Step 4: Cancel out common factors from the numerator and denominator. (m-n)(m+n)(m-n)(m-n) × -n(m-n)m(m+n) (m+n)/(m-n) × (-n)/(m(m+n)) Now, cancel (m+n) from the numerator of the first term and the denominator of the second term. (m+n)m-n × (-n)/(m(m+n)) Step 5: Multiply the remaining terms. (1)/(m-n) × (-n)/(m) = (-n)/(m(m-n)) The simplified expression is: (-n)/(m(m-n))