Question
\( 3^{x+1}=9^{x}.\frac{1}{27} \)
Ask by Norris Powers. in South Africa
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( x = 4 \)
Solution
To solve the equation:
\[
3^{x+1} = 9^{x} \cdot \frac{1}{27}
\]
Let's follow the steps below:
### Step 1: Express All Terms with the Same Base
First, notice that all numbers can be expressed as powers of 3:
- \( 9 = 3^2 \)
- \( 27 = 3^3 \)
Rewrite the equation using these expressions:
\[
3^{x+1} = (3^2)^x \cdot 3^{-3}
\]
### Step 2: Simplify the Exponents
Apply the power rule \((a^m)^n = a^{mn}\):
\[
3^{x+1} = 3^{2x} \cdot 3^{-3} = 3^{2x - 3}
\]
Now the equation is:
\[
3^{x+1} = 3^{2x - 3}
\]
### Step 3: Set the Exponents Equal
Since the bases are the same and the equation holds, the exponents must be equal:
\[
x + 1 = 2x - 3
\]
### Step 4: Solve for \( x \)
Subtract \( x \) from both sides:
\[
1 = x - 3
\]
Add 3 to both sides:
\[
x = 4
\]
### Conclusion
The solution to the equation is:
\[
x = 4
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
To solve the equation \( 3^{x+1}=9^{x} \cdot \frac{1}{27} \), we can start by rewriting everything in terms of the base 3. We know that \( 9 \) can be expressed as \( 3^2 \) and \( 27 \) can be expressed as \( 3^3 \). Thus, we have: \[ 9^x = (3^2)^x = 3^{2x} \] \[ \frac{1}{27} = 3^{-3} \] Now we can rewrite the original equation: \[ 3^{x+1} = 3^{2x} \cdot 3^{-3} \] Now, we can combine the right-hand side: \[ 3^{x+1} = 3^{2x - 3} \] Since the bases are the same, we can set the exponents equal to each other: \[ x + 1 = 2x - 3 \] Now, let's solve for \( x \): \[ 1 + 3 = 2x - x \] \[ 4 = x \] So, the solution to the equation is \( x = 4 \).
