Pregunta
Solve the equation. Simplify the answer as much as possible. \[ 2^{2 a+8}=\left(\frac{1}{2}\right)^{a-2} \]
Ask by Campos Pope. in the United States
Nov 13,2024
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
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\(a=-2\)
Solución
Solve the equation \( 2^{2a+8}=(\frac{1}{2})^{a-2} \).
Solve the equation by following steps:
- step0: Solve for \(a\):
\(2^{2a+8}=\left(\frac{1}{2}\right)^{a-2}\)
- step1: Rewrite the expression:
\(2^{2a+8}=2^{-a+2}\)
- step2: Set the exponents equal:
\(2a+8=-a+2\)
- step3: Move the variable to the left side:
\(2a+8+a=2\)
- step4: Add the terms:
\(3a+8=2\)
- step5: Move the constant to the right side:
\(3a=2-8\)
- step6: Subtract the numbers:
\(3a=-6\)
- step7: Divide both sides:
\(\frac{3a}{3}=\frac{-6}{3}\)
- step8: Divide the numbers:
\(a=-2\)
The solution to the equation \(2^{2a+8}=(\frac{1}{2})^{a-2}\) is \(a=-2\).
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Simplifique esta solución Beyond the Answer
To solve the equation \[ 2^{2a+8} = \left(\frac{1}{2}\right)^{a-2}, \] we first rewrite \(\frac{1}{2}\) as \(2^{-1}\): \[ \left(\frac{1}{2}\right)^{a-2} = (2^{-1})^{a-2} = 2^{-(a-2)} = 2^{-a + 2}. \] Now we can rewrite the equation: \[ 2^{2a + 8} = 2^{-a + 2}. \] Since the bases are the same, we can set the exponents equal to each other: \[ 2a + 8 = -a + 2. \] Now, we solve for \(a\). First, add \(a\) to both sides: \[ 2a + a + 8 = 2, \] which simplifies to: \[ 3a + 8 = 2. \] Next, subtract 8 from both sides: \[ 3a = 2 - 8, \] or \[ 3a = -6. \] Now, divide both sides by 3: \[ a = -2. \] Thus, the simplified answer is \[ \boxed{-2}. \]
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