Question
Real Tutor Solution
Tutor-Verified Answer
Answer
The simplified form of the expression is \(2^x - 1\).
Solution
To simplify the expression:
\[
\frac{2^{2x} - 1}{2^x + 1}
\]
**Step 1: Recognize the Structure**
Notice that \(2^{2x}\) can be written as \((2^x)^2\). This allows us to see the numerator as a difference of squares:
\[
2^{2x} - 1 = (2^x)^2 - 1^2
\]
**Step 2: Factor the Numerator**
Using the difference of squares formula, \(a^2 - b^2 = (a - b)(a + b)\), we can factor the numerator:
\[
(2^x)^2 - 1^2 = (2^x - 1)(2^x + 1)
\]
**Step 3: Simplify the Expression**
Substitute the factored form back into the original expression:
\[
\frac{(2^x - 1)(2^x + 1)}{2^x + 1}
\]
Cancel out the common factor \((2^x + 1)\) in the numerator and the denominator:
\[
\frac{(2^x - 1)\cancel{(2^x + 1)}}{\cancel{2^x + 1}} = 2^x - 1
\]
**Final Simplified Form:**
\[
2^x - 1
\]
**Conclusion:**
The simplified form of the expression \(\frac{2^{2x} - 1}{2^x + 1}\) is:
\[
2^x - 1
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
To tackle the expression \( \frac{2^{2x}-1}{2^{x}+1} \), you can spot that \( 2^{2x} - 1 \) is a difference of squares! It can be factored as \( (2^x - 1)(2^x + 1) \). This nifty trick simplifies your work! So now, you can rewrite the entire expression as: \[ \frac{(2^x - 1)(2^x + 1)}{2^x + 1} \] As long as \( 2^x + 1 \neq 0 \), you can cancel it out, leading to: \[ 2^x - 1 \] This clever simplification turns a complicated-looking expression into something much easier to handle! But don’t be too hasty—make sure to note that the cancellation is valid only where \( 2^x + 1 \neq 0 \). Given that \( 2^x + 1 \) is always positive for real \( x \), you're good to go!
