Pregunta
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The expression simplifies to 0.
Solución
Let's simplify the given expression:
\[
\frac{3^{x} - 3^{x}}{3^{x} + 3^{x+2}}
\]
**Step 1: Simplify the Numerator**
\[
3^{x} - 3^{x} = 0
\]
**Step 2: Simplify the Denominator**
\[
3^{x} + 3^{x+2} = 3^{x} + 3^{x} \cdot 3^{2} = 3^{x} + 9 \cdot 3^{x} = (1 + 9) \cdot 3^{x} = 10 \cdot 3^{x}
\]
**Step 3: Combine Numerator and Denominator**
\[
\frac{0}{10 \cdot 3^{x}} = 0
\]
Since the denominator \(10 \cdot 3^{x}\) is never zero for any real value of \(x\) (because \(3^{x} > 0\) for all real \(x\)), the entire expression simplifies to:
\[
0
\]
**Final Answer:**
0
Revisado y aprobado por el equipo de tutoría de UpStudy
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Simplifique esta solución Mind Expander
First, let’s simplify the expression step by step. The numerator simplifies like this: \( 3^{x} - 3^{x} = 0 \). So, the entire expression becomes \( \frac{0}{3^{x} + 3^{x + 2}} \). Since the numerator is 0, regardless of what the denominator turns out to be (as long as it’s not zero), the whole expression evaluates to 0. Feeling confused by negative powers? Remember that \( 3^{x + 2} \) can be rewritten as \( 3^x \cdot 3^2 \) which equals \( 9 \cdot 3^x \). Now you can see that the denominator is always positive for real \( x \), confirming that this expression is indeed 0 for all \( x \).
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