Question
Upstudy AI Solution
Tutor-Verified Answer
Answer
A solução é \( x = \frac{4}{7} \).
Solution
**Passo 1:** Escrever as bases \(25\) e \(125\) em termos de \(5\):
\[
25 = 5^2 \quad \text{e} \quad 125 = 5^3.
\]
**Passo 2:** Reescrever a equação substituindo as bases:
\[
(5^2)^{2x+1} = (5^3)^{-x+2}.
\]
**Passo 3:** Aplicar a propriedade das potências \((a^b)^c = a^{b \cdot c}\):
\[
5^{2(2x+1)} = 5^{3(-x+2)}.
\]
**Passo 4:** Simplificar os expoentes:
\[
5^{4x+2} = 5^{-3x+6}.
\]
**Passo 5:** Como as bases são iguais, igualamos os expoentes:
\[
4x + 2 = -3x + 6.
\]
**Passo 6:** Resolver a equação:
\[
4x + 3x = 6 - 2 \quad \Rightarrow \quad 7x = 4 \quad \Rightarrow \quad x = \frac{4}{7}.
\]
Portanto, a solução é:
\[
x = \frac{4}{7}.
\]
Answered by UpStudy AI and reviewed by a Professional Tutor
Like
error msg
Explain 
Simplify this solution Beyond the Answer
To solve the equation \( 25^{2x+1} = 125^{-x+2} \), we can express both sides with a common base. We know that \( 25 = 5^2 \) and \( 125 = 5^3 \). Therefore, we can rewrite the equation as: \[ (5^2)^{2x+1} = (5^3)^{-x+2} \] This simplifies to: \[ 5^{4x+2} = 5^{-3x+6} \] Since the bases are equal, we can set the exponents equal to each other: \[ 4x + 2 = -3x + 6 \] Now, let's solve for \(x\): \[ 4x + 3x = 6 - 2 \] \[ 7x = 4 \] \[ x = \frac{4}{7} \] So the solution for the equation is: \[ \boxed{\frac{4}{7}} \]
