Pregunta
upstudy study bank question image url

5. Match the exponent equality to the correct solution for \( x \). \( 16^{x+2}=4^{x-2} \) 3. \( 25^{x-5}=5 \) 4. \( 27^{x+3}=5^{x+2} \)

Ask by Reese Pope. in the United States
Mar 14,2025

Solución de inteligencia artificial de Upstudy

Respuesta verificada por el tutor

Responder

The solutions are: 1. \( x = -6 \) 2. \( x = \frac{11}{2} \) 3. \( x = \frac{2\ln 5 - 9\ln 3}{3\ln 3 - \ln 5} \)

Solución

**Problem 5.1: \(16^{x+2}=4^{x-2}\)** 1. Express 16 and 4 as powers of 2: \[ 16 = 2^4 \quad \text{and} \quad 4 = 2^2 \] 2. Rewrite the equation: \[ (2^4)^{x+2} = (2^2)^{x-2} \quad \Longrightarrow \quad 2^{4(x+2)} = 2^{2(x-2)} \] 3. Equate the exponents: \[ 4(x+2) = 2(x-2) \] 4. Solve the linear equation: \[ 4x + 8 = 2x - 4 \quad \Longrightarrow \quad 4x - 2x = -4 - 8 \quad \Longrightarrow \quad 2x = -12 \] \[ x = -6 \] --- **Problem 5.2: \(25^{x-5}=5\)** 1. Express 25 as a power of 5: \[ 25 = 5^2 \] 2. Rewrite the equation: \[ (5^2)^{x-5} = 5 \quad \Longrightarrow \quad 5^{2(x-5)} = 5^1 \] 3. Equate the exponents: \[ 2(x-5) = 1 \] 4. Solve the linear equation: \[ 2x - 10 = 1 \quad \Longrightarrow \quad 2x = 11 \] \[ x = \frac{11}{2} \] --- **Problem 5.3: \(27^{x+3}=5^{x+2}\)** 1. Express 27 as a power of 3: \[ 27 = 3^3 \] 2. Rewrite the equation: \[ (3^3)^{x+3} = 5^{x+2} \quad \Longrightarrow \quad 3^{3(x+3)} = 5^{x+2} \] 3. Take the natural logarithm of both sides: \[ \ln\left(3^{3(x+3)}\right) = \ln\left(5^{x+2}\right) \] 4. Use the power rule of logarithms: \[ 3(x+3)\ln 3 = (x+2)\ln 5 \] 5. Distribute and collect like terms: \[ 3x\ln 3 + 9\ln 3 = x\ln 5 + 2\ln 5 \] \[ 3x\ln 3 - x\ln 5 = 2\ln 5 - 9\ln 3 \] 6. Factor out \( x \): \[ x(3\ln 3 - \ln 5) = 2\ln 5 - 9\ln 3 \] 7. Solve for \( x \): \[ x = \frac{2\ln 5 - 9\ln 3}{3\ln 3 - \ln 5} \] --- **Final Answers:** 1. For \( 16^{x+2}=4^{x-2} \), \( x = -6 \). 2. For \( 25^{x-5}=5 \), \( x = \frac{11}{2} \). 3. For \( 27^{x+3}=5^{x+2} \), \( x = \frac{2\ln 5 - 9\ln 3}{3\ln 3 - \ln 5} \).

Respondido por UpStudy AI y revisado por un tutor profesional

error msg
Explicar
Simplifique esta solución

Beyond the Answer

To solve \( 16^{x+2}=4^{x-2} \), we can express both sides as powers of 2: \( 16 = 2^4 \) and \( 4 = 2^2 \). Thus, the equation becomes \( (2^4)^{x+2} = (2^2)^{x-2} \) or \( 2^{4(x+2)} = 2^{2(x-2)} \). This simplifies to \( 4(x+2) = 2(x-2) \). Solving this gives \( x = -3 \). Next, for \( 25^{x-5}=5 \), we rewrite \( 25 \) as \( 5^2 \), resulting in \( (5^2)^{x-5}=5^1 \). Therefore, \( 5^{2(x-5)} = 5^1 \) leads to \( 2(x-5) = 1 \), yielding \( x = 6.5 \) or \( x = \frac{13}{2} \). Finally, we need to examine \( 27^{x+3}=5^{x+2} \) by rewriting \( 27 \) as \( 3^3 \). Hence, \( (3^3)^{x+3}=5^{x+2} \) gives \( 3^{3(x+3)} = 5^{x+2} \), which doesn't yield a product of simple base equality. Therefore, this equation generally requires logs or numerical methods for solutions. Have fun exploring the power of exponents! They pop up everywhere, from simplifying equations to calculating compounds in finance – like determining interest over time, showing just how every little exponent can multiply your returns!

preguntas relacionadas

¡Prueba Premium ahora!
¡Prueba Premium y hazle a Thoth AI preguntas de matemáticas ilimitadas ahora!
Quizas mas tarde Hazte Premium
Estudiar puede ser una verdadera lucha
¿Por qué no estudiarlo en UpStudy?
Seleccione su plan a continuación
Prima

Puedes disfrutar

Empieza ahora
  • Explicaciones paso a paso
  • Tutores expertos en vivo 24/7
  • Número ilimitado de preguntas
  • Sin interrupciones
  • Acceso completo a Respuesta y Solución
  • Acceso completo al chat de PDF, al chat de UpStudy y al chat de navegación
Básico

Totalmente gratis pero limitado

  • Solución limitada
Bienvenido a ¡Estudia ahora!
Inicie sesión para continuar con el recorrido de Thoth AI Chat
Continuar con correo electrónico
O continuar con
Al hacer clic en "Iniciar sesión", acepta nuestros términos y condiciones. Términos de Uso & Política de privacidad