Question
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Tutor-Verified Answer
Answer
The simplified form is \( -4 - 2^{x} \).
Solution
Calculate or simplify the expression \( (16-4^{x})/(2^{x}-4) \).
Simplify the expression by following steps:
- step0: Solution:
\(\frac{\left(16-4^{x}\right)}{\left(2^{x}-4\right)}\)
- step1: Remove the parentheses:
\(\frac{16-4^{x}}{2^{x}-4}\)
- step2: Evaluate:
\(\frac{4\left(2^{2}-2^{2x-2}\right)}{4\left(2^{x-2}-1\right)}\)
- step3: Evaluate:
\(\frac{2^{2}-2^{2x-2}}{2^{x-2}-1}\)
- step4: Calculate:
\(\frac{4-2^{2x-2}}{2^{x-2}-1}\)
- step5: Rewrite the expression:
\(\frac{4-\frac{1}{4}\left(2^{x}\right)^{2}}{\frac{1}{4}\times 2^{x}-1}\)
- step6: Rewrite the expression:
\(\frac{\frac{16-\left(2^{x}\right)^{2}}{4}}{\frac{1}{4}\times 2^{x}-1}\)
- step7: Rewrite the expression:
\(\frac{\frac{16-\left(2^{x}\right)^{2}}{4}}{\frac{2^{x}-4}{4}}\)
- step8: Multiply by the reciprocal:
\(\frac{16-\left(2^{x}\right)^{2}}{4}\times \frac{4}{2^{x}-4}\)
- step9: Rewrite the expression:
\(\frac{\left(2^{x}-4\right)\left(-4-2^{x}\right)}{4}\times \frac{4}{2^{x}-4}\)
- step10: Reduce the fraction:
\(\left(-4-2^{x}\right)\times 1\)
- step11: Multiply the terms:
\(-\left(4+2^{x}\right)\)
- step12: Calculate:
\(-4-2^{x}\)
The simplified form of the expression \( \frac{16-4^{x}}{2^{x}-4} \) is \( -4-2^{x} \).
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Mind Expander
The expression \( \frac{16-4^{x}}{2^{x}-4} \) can be simplified by recognizing that \( 4^x = (2^2)^x = 2^{2x} \). Thus, the expression becomes \( \frac{16 - 2^{2x}}{2^x - 4} \). You can further simplify the numerator \( 16 - 2^{2x} \) to \( 4 - (2^x)^2 \), which is a difference of squares, hence it can be factored as \( (4 - 2^x)(4 + 2^x) \). Now, rewriting the original expression, we have: \[ \frac{(4 - 2^x)(4 + 2^x)}{2^x - 4} \] Notice that \( 2^x - 4 \) is equivalent to \( -(4 - 2^x) \). Thus, the expression simplifies into: \[ -(4 + 2^x) \] So, the final simplified result of the expression is \( -(4 + 2^x) \).
