Question

select the correct answer from each drop-down menu. the simplest form of the expression \\(\frac{4m - 17}{m^2 - 16} + \frac{3m - 11}{m^2 - 16}\\) has \\(\square\\) in the numerator and \\(\square\\) in the denominator

Explanation:

Step1: Add the numerators

Since the denominators are the same ($m^2 - 16$), we add the numerators: $(4m - 17)+(3m - 11)=4m+3m-17 - 11 = 7m-28$.

Step2: Simplify the numerator

Factor the numerator: $7m - 28=7(m - 4)$.

Step3: Factor the denominator

Factor the denominator $m^2 - 16$ using the difference of squares: $m^2 - 16=(m + 4)(m - 4)$.

Step4: Simplify the fraction

Cancel out the common factor $(m - 4)$ from the numerator and the denominator: $\frac{7(m - 4)}{(m + 4)(m - 4)}=\frac{7}{m + 4}$. Wait, no, wait, let's re - check the addition of numerators. Wait, $(4m-17)+(3m - 11)=4m+3m-17 - 11=7m - 28$. Then factor numerator: $7m-28 = 7(m - 4)$. Denominator: $m^2-16=(m - 4)(m + 4)$. So when we add the two fractions $\frac{4m - 17}{m^2 - 16}+\frac{3m - 11}{m^2 - 16}=\frac{(4m - 17)+(3m - 11)}{m^2 - 16}=\frac{7m-28}{m^2 - 16}$. Then simplify numerator and denominator: numerator $7m - 28 = 7(m - 4)$, denominator $(m - 4)(m + 4)$. So the simplified form is $\frac{7(m - 4)}{(m - 4)(m + 4)}=\frac{7}{m + 4}$ (assuming $m
eq4$). But the question is about the numerator and denominator of the simplified form? Wait, no, maybe the question is about the numerator after adding and before simplifying? Wait, the original expression is adding two fractions with the same denominator. So first, add the numerators: $(4m - 17)+(3m - 11)=7m-28$. The denominator remains $m^2 - 16$ before simplification. But if we simplify, numerator is 7 and denominator is $m + 4$. Wait, maybe the problem is to find the numerator and denominator of the sum before canceling? Wait, let's re - do the addition:

$\frac{4m-17}{m^2 - 16}+\frac{3m - 11}{m^2 - 16}=\frac{(4m-17)+(3m - 11)}{m^2 - 16}=\frac{4m+3m-17 - 11}{m^2 - 16}=\frac{7m-28}{m^2 - 16}$

So the numerator is $7m - 28$ (or $7(m - 4)$) and the denominator is $m^2 - 16$ (or $(m + 4)(m - 4)$) before simplification. If we simplify, numerator is 7 and denominator is $m + 4$.

Wait, maybe I made a mistake in the addition. Let's recalculate the numerator:

$4m-17+3m - 11=(4m + 3m)+(-17-11)=7m-28$. Yes, that's correct.

So the sum of the two fractions is $\frac{7m - 28}{m^2 - 16}$. So the numerator is $7m - 28$ (or $7(m - 4)$) and the denominator is $m^2 - 16$ (or $(m - 4)(m + 4)$). If we simplify by canceling $(m - 4)$, the numerator becomes 7 and the denominator becomes $m + 4$.

Answer:

The numerator (after adding and before simplifying) is $7m - 28$ (or $7(m - 4)$) and the denominator is $m^2 - 16$ (or $(m - 4)(m + 4)$). If we consider the simplified form, the numerator is $7$ and the denominator is $m + 4$.