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b) \( 7^{x^{2}-9}=1 \)

Ask by Patel Bernard. in Brazil
Mar 20,2025

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Tutor-Verified Answer

Answer

\( x = 3 \) ou \( x = -3 \).

Solution

**Passo 1:** Observe que uma potência da forma \( a^k \) é igual a 1 se, e somente se, \( k = 0 \) (levando em conta que \( a \) é um número real positivo diferente de 1). No caso, temos: \[ 7^{x^{2} - 9} = 1 \] Como \( 7 \neq 1 \), concluímos que: \[ x^{2} - 9 = 0 \] **Passo 2:** Resolvendo a equação: \[ x^{2} - 9 = 0 \quad \Rightarrow \quad x^{2} = 9 \] Tomando a raiz quadrada de ambos os lados, considerando as duas possibilidades (positiva e negativa): \[ x = 3 \quad \text{ou} \quad x = -3 \] **Resposta final:** \( x = 3 \) ou \( x = -3 \).

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To solve the equation \( 7^{x^{2}-9}=1 \), we recognize that any number raised to the power of 0 is equal to 1. Therefore, we can set the exponent equal to 0: \[ x^{2} - 9 = 0 \] Solving this gives: \[ x^{2} = 9 \] Taking the square root of both sides yields: \[ x = 3 \quad \text{or} \quad x = -3 \] Now we have two solutions: \( x=3 \) and \( x=-3 \). In summary, the solutions to the equation \( 7^{x^{2}-9}=1 \) are \( x = 3 \) and \( x = -3 \).

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