Question
\( 1.1(x+9)\left(3^{x-2}-27\right)=0 \)
Ask by Wagner Lane. in South Africa
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The solutions are \( x = -9 \) and \( x = 5 \).
Solution
To solve the equation:
\[
1.1(x + 9)\left(3^{x - 2} - 27\right) = 0
\]
We can use the **Zero Product Property**, which states that if a product of factors equals zero, then at least one of the factors must be zero. Let's break down the equation into its factors:
1. **First Factor:** \(1.1\)
Since \(1.1 \neq 0\), this factor does not provide any solutions.
2. **Second Factor:** \(x + 9\)
Setting this equal to zero:
\[
x + 9 = 0 \implies x = -9
\]
3. **Third Factor:** \(3^{x - 2} - 27\)
Setting this equal to zero:
\[
3^{x - 2} - 27 = 0 \implies 3^{x - 2} = 27
\]
Recall that \(27\) can be expressed as a power of \(3\):
\[
27 = 3^3
\]
So,
\[
3^{x - 2} = 3^3
\]
Since the bases are the same, we can set the exponents equal to each other:
\[
x - 2 = 3 \implies x = 5
\]
**Final Solutions:**
\[
x = -9 \quad \text{and} \quad x = 5
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Bonus Knowledge
To solve the equation \( 1.1(x+9)(3^{x-2}-27)=0 \), we can set each factor to zero. First, from \( 1.1(x+9) = 0 \), we find that \( x + 9 = 0 \), which gives us \( x = -9 \). Next, we look at \( 3^{x-2} - 27 = 0 \). We rewrite 27 as \( 3^3 \): \[ 3^{x-2} = 3^3 \] Since the bases are the same, we equate the exponents: \[ x - 2 = 3 \] Solving for \( x \), we get \( x = 5 \). Thus, the solutions to the equation are \( x = -9 \) and \( x = 5 \).
