(5m)/(m^(2)-24 mn+144n^(2))+(2n)/(m^(2)-144n^(2)) Which expression is equivalent to the sum? Choose 1 answer: A) (5m^(2)+60 mn+2n^(2))/((m-12 n)(m+12 n)) B) (5m^(2)+62 mn-24n^(2))/((m-12 n)^(2)(m+12 n)) C) (5m-24 mn+2n)/((m-12 n)(m-12 n))

Question

(5m)/(m^(2)-24 mn+144n^(2))+(2n)/(m^(2)-144n^(2)) Which expression is equivalent to the sum? Choose 1 answer: A)  (5m^(2)+60 mn+2n^(2))/((m-12 n)(m+12 n)) B)  (5m^(2)+62 mn-24n^(2))/((m-12 n)^(2)(m+12 n)) C)  (5m-24 mn+2n)/((m-12 n)(m-12 n))

Bytelearn · Accepted Answer

Identify Denominators and Factor: First, let's identify the denominators of the two fractions and factor them if possible. The first denominator is $m^2 - 24mn + 144n^2$, which is a perfect square trinomial and can be factored as $(m - 12n)^2$. The second denominator is $m^2 - 144n^2$, which is a difference of squares and can be factored as $(m - 12n)(m + 12n)$.Find Common Denominator: Now, let's find a common denominator for the two fractions. The least common denominator (LCD) is the product of the distinct factors, which in this case is $(m - 12n)(m + 12n)$.Express Fractions with LCD: Next, we need to express each fraction with the common denominator. For the first fraction, the denominator is already $(m - 12n)^2$, which is missing the factor $(m + 12n)$. So we multiply the numerator and denominator of the first fraction by $(m + 12n)$. For the second fraction, the denominator is $(m - 12n)(m + 12n)$, which is already the LCD, so we don't need to change it.Combine Numerators: After adjusting the fractions, we have: $$\frac{5m(m + 12n)}{(m - 12n)(m + 12n)} + \frac{2n}{(m - 12n)(m + 12n)}$$Distribute and Combine Terms: Now, let's combine the numerators over the common denominator: $$\frac{5m(m + 12n) + 2n}{(m - 12n)(m + 12n)}$$Final Combined Fraction: We distribute the $5m$ across the terms in the parentheses and combine like terms in the numerator: $$5m(m) + 5m(12n) + 2n = 5m^2 + 60mn + 2n$$Check Answer Choices: Now we have the combined fraction: $$\frac{5m^2 + 60mn + 2n}{(m - 12n)(m + 12n)}$$We look at the answer choices to see which one matches our result. The correct answer is: $$A) \frac{5m^2 + 60mn + 2n^2}{(m - 12n)(m + 12n)}$$ However, we made a mistake in the previous step by not squaring the $n$ in the term $2n$. The correct term should be $2n^2$.

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