Question
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Tutor-Verified Answer
Answer
The simplified form is \( 2^{x} - 3 \).
Solution
Calculate or simplify the expression \( (2^(2x)-9)/(2^x+3) \).
Simplify the expression by following steps:
- step0: Solution:
\(\frac{\left(2^{2x}-9\right)}{\left(2^{x}+3\right)}\)
- step1: Remove the parentheses:
\(\frac{2^{2x}-9}{2^{x}+3}\)
- step2: Rewrite the expression:
\(\frac{\left(2^{x}\right)^{2}-9}{2^{x}+3}\)
- step3: Calculate:
\(\frac{\left(2^{x}-3\right)\left(2^{x}+3\right)}{2^{x}+3}\)
- step4: Reduce the fraction:
\(2^{x}-3\)
The simplified form of the expression \( \frac{2^{2x}-9}{2^{x}+3} \) is \( 2^{x}-3 \).
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Simplify this solution Beyond the Answer
Sure! Let’s dive a bit deeper into this expression. First, we can rewrite the numerator \(2^{2x}\) as \((2^x)^2\). So now the expression looks like \(\frac{(2^x)^2 - 9}{2^x + 3}\). This resembles the difference of squares, which is \(a^2 - b^2 = (a - b)(a + b)\), where in our case \(a = 2^x\) and \(b = 3\). This means we can factor the numerator as \((2^x - 3)(2^x + 3)\). Next, we simplify! The denominator is \(2^x + 3\), which we can cancel with the factor in the numerator. This leaves us with the simplified expression: \(2^x - 3\), assuming \(2^x \neq -3\) (which is always true since \(2^x\) is positive). Voilà! Now you've got a much simpler expression to work with!
