Question
Simplify the foliowing: (a) \( \frac{2^{2 x}-2^{x}}{2^{x}-1} \)
Ask by Mcfarlane Wagner. in South Africa
Jan 20,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The simplified form of the expression is \(2^{x}\).
Solution
To simplify the expression:
\[
\frac{2^{2x} - 2^{x}}{2^{x} - 1}
\]
Follow these steps:
1. **Rewrite the Numerator:**
Notice that \(2^{2x}\) can be written as \((2^{x})^2\). Therefore, the numerator becomes:
\[
(2^{x})^2 - 2^{x} = 2^{x}(2^{x} - 1)
\]
2. **Simplify the Fraction:**
Substitute the rewritten numerator back into the original expression:
\[
\frac{2^{x}(2^{x} - 1)}{2^{x} - 1}
\]
The \((2^{x} - 1)\) terms in the numerator and denominator cancel each other out (assuming \(2^{x} - 1 \neq 0\)), leaving:
\[
2^{x}
\]
**Final Simplified Form:**
\[
2^{x}
\]
**Note:** The simplification holds as long as \(2^{x} - 1 \neq 0\), which implies \(x \neq 0\). At \(x = 0\), the original expression is undefined (results in a \(\frac{0}{0}\) form), whereas the simplified expression \(2^{x}\) equals 1.
**Answer:**
After simplifying, the expression is equal to \(2^{\,x}\).
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To simplify \( \frac{2^{2x}-2^{x}}{2^{x}-1} \), we can use substitution. Let \( y = 2^{x} \). Then, \( 2^{2x} = (2^{x})^2 = y^2 \). Substituting these into the expression gives: \[ \frac{y^2 - y}{y - 1} \] Now, factor the numerator: \[ y^2 - y = y(y - 1) \] So our expression becomes: \[ \frac{y(y - 1)}{y - 1} \] Assuming \( y - 1 \neq 0 \) (or \( 2^{x} \neq 1 \)), we can cancel \( y - 1 \): \[ y = 2^{x} \] Thus, the simplified expression is: \[ 2^{x} \quad \text{for } 2^{x} \neq 1 \]
