Question
(a) \( A=\frac{3^{n}-4}{2^{n} \cdot 3^{n}-2^{n} \cdot 2^{2}}= \)
Ask by Spencer Ward. in South Africa
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( A = \frac{1}{2^{n}} \)
Solution
Let's simplify the expression:
\[
A = \frac{3^{n} - 4}{2^{n} \cdot 3^{n} - 2^{n} \cdot 2^{2}}
\]
**Step 1: Factor the Denominator**
Notice that both terms in the denominator have a common factor of \(2^{n}\):
\[
2^{n} \cdot 3^{n} - 2^{n} \cdot 2^{2} = 2^{n} (3^{n} - 4)
\]
**Step 2: Simplify the Expression**
Now, substitute the factored form of the denominator back into the expression for \(A\):
\[
A = \frac{3^{n} - 4}{2^{n} (3^{n} - 4)}
\]
Assuming \(3^{n} \neq 4\), we can cancel out the common factor \((3^{n} - 4)\) from the numerator and the denominator:
\[
A = \frac{1}{2^{n}}
\]
**Final Answer:**
\[
A = \frac{1}{2^{n}} \quad \text{or equivalently} \quad A = 2^{-n}
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
To simplify the expression \( A=\frac{3^{n}-4}{2^{n} \cdot 3^{n}-2^{n} \cdot 2^{2}} \), let's start with the denominator: \[ 2^{n} \cdot 3^{n} - 2^{n} \cdot 2^{2} = 2^{n} \cdot (3^{n} - 4) \] Now we can rewrite \( A \): \[ A = \frac{3^{n} - 4}{2^{n} \cdot (3^{n} - 4)} \] Assuming \( 3^{n} - 4 \neq 0 \) (to avoid division by zero), we can simplify this to: \[ A = \frac{1}{2^{n}} \] Thus, the final simplified expression for \( A \) is: \[ A = \frac{1}{2^{n}} \]
