Question
Real Tutor Solution
Tutor-Verified Answer
Answer
\( x = -3 \)
Solution
Solve the equation \( 3^{x} \times 3^{-3}=9^{x} \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(3^{x}\times 3^{-3}=9^{x}\)
- step1: Calculate:
\(3^{x-3}=9^{x}\)
- step2: Rewrite the expression:
\(3^{x-3}=3^{2x}\)
- step3: Set the exponents equal:
\(x-3=2x\)
- step4: Move the variable to the left side:
\(x-3-2x=0\)
- step5: Subtract the terms:
\(-x-3=0\)
- step6: Move the constant to the right side:
\(-x=0+3\)
- step7: Remove 0:
\(-x=3\)
- step8: Change the signs:
\(x=-3\)
The solution to the equation \(3^{x} \times 3^{-3} = 9^{x}\) is \(x = -3\).
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Simplify this solution Beyond the Answer
To solve the equation \( \frac{3^{x} \times 3^{-3}}{9^{x}}=1 \), we can first express \( 9^{x} \) in terms of base 3. Since \( 9 = 3^{2} \), we have \( 9^{x} = (3^{2})^{x} = 3^{2x} \). Now substitute this into the equation: \[ \frac{3^{x} \times 3^{-3}}{3^{2x}}=1 \] This simplifies to: \[ \frac{3^{x - 3}}{3^{2x}} = 1 \] Using the properties of exponents, we can combine the bases: \[ 3^{x - 3 - 2x} = 1 \] This further simplifies to: \[ 3^{-x - 3} = 1 \] Since \( 3^{0} = 1 \), we can set the exponent equal to 0: \[ -x - 3 = 0 \] Solving for \( x \), we get: \[ -x = 3 \quad \Rightarrow \quad x = -3 \] So the solution is \( x = -3 \).
