Pregunta
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Respuesta verificada por el tutor
Responder
\[
\frac{8(t - 1)}{t^2 - 16}
\]
Solución
To simplify the expression:
\[
\frac{5}{t+4} + \frac{3}{t-4}
\]
**Step 1: Find a Common Denominator**
The denominators are \( t + 4 \) and \( t - 4 \). The least common denominator (LCD) is:
\[
(t + 4)(t - 4) = t^2 - 16
\]
**Step 2: Rewrite Each Fraction with the LCD**
\[
\frac{5}{t+4} = \frac{5(t - 4)}{(t + 4)(t - 4)} = \frac{5(t - 4)}{t^2 - 16}
\]
\[
\frac{3}{t-4} = \frac{3(t + 4)}{(t - 4)(t + 4)} = \frac{3(t + 4)}{t^2 - 16}
\]
**Step 3: Combine the Fractions**
\[
\frac{5(t - 4) + 3(t + 4)}{t^2 - 16}
\]
**Step 4: Simplify the Numerator**
\[
5(t - 4) + 3(t + 4) = 5t - 20 + 3t + 12 = 8t - 8
\]
**Step 5: Factor the Numerator**
\[
8t - 8 = 8(t - 1)
\]
**Final Simplified Expression**
\[
\frac{8(t - 1)}{t^2 - 16}
\]
or, factored further,
\[
\frac{8(t - 1)}{(t + 4)(t - 4)}
\]
**Summary:**
\[
\frac{5}{t+4} + \frac{3}{t-4} = \frac{8(t - 1)}{t^2 - 16}
\]
Respondido por UpStudy AI y revisado por un tutor profesional
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Simplifique esta solución The Deep Dive
To add these fractions, you'll first need a common denominator, which is \((t + 4)(t - 4)\). Rewrite each fraction accordingly: \[ \frac{5(t-4)}{(t+4)(t-4)} + \frac{3(t+4)}{(t+4)(t-4)} \] Now, combine the numerators: \[ \frac{5(t-4) + 3(t+4)}{(t+4)(t-4)} \] Simplifying the numerator: \[ 5t - 20 + 3t + 12 = 8t - 8 \] So, we have: \[ \frac{8t - 8}{(t + 4)(t - 4)} \] You can now factor out the 8 from the numerator: \[ \frac{8(t - 1)}{(t + 4)(t - 4)} \] This is the simplified expression!
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