Question Statement:
If a real variable $x$ satisfies $3^{x^2} = 27 \times 9^x$, then the value of $\frac{2^{x^2}}{(2^x)^2}$ is:
Options:
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(A) $2^{-1}$
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(B) $2^0$
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(C) $2^3$
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(D) $2^{15}$
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2025 Q.9)**
**1. Intuitive Physical Setup & Given Parameters**:
We are given that real variable $x$ satisfies:
$3^{x^2} = 27 \times 9^x$
We need to find the value of:
$E = \frac{2^{x^2}}{(2^x)^2}$
**2. Step-by-Step Mathematical Derivation**:
1. **Solve for $x$ from the Base-3 Equation**:
Express all terms in base 3:
$27 = 3^3, \quad 9^x = (3^2)^x = 3^{2x}$
$3^{x^2} = 3^3 \times 3^{2x} = 3^{x^2} = 3^{2x + 3}$
Equating exponents:
$x^2 = 2x + 3$
$x^2 - 2x - 3 = 0$
$(x - 3)(x + 1) = 0 \implies x = 3 \quad \text{or} \quad x = -1$
2. **Evaluate the Target Expression $E$**:
Simplify $E$ using exponent laws:
$E = \frac{2^{x^2}}{(2^x)^2} = \frac{2^{x^2}}{2^{2x}} = 2^{x^2 - 2x}$
3. **Substitute $x^2 - 2x = 3$**:
From the quadratic equation above, notice that:
$x^2 - 2x = 3$
for both roots ($x = 3$ and $x = -1$).
Therefore:
$E = 2^{x^2 - 2x} = 2^3$
**3. Comprehensive Options & Distractor Analysis**:
- **(A) 2^-1**: ❌ **INCORRECT**. Result of algebraic sign errors.
- **(B) 2^0**: ❌ **INCORRECT**. Occurs if exponents are mistakenly canceled to 0.
- **(C) 2^3**: ✅ **CORRECT**. Since $x^2 - 2x = 3$, $2^{x^2 - 2x} = 2^3$.
- **(D) 2^15**: ❌ **INCORRECT**. Exponent multiplication error.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **C**
- **Exam Takeaway**: Notice algebraic relationships before computing roots directly: $x^2 - 2x = 3$ directly transforms the denominator exponent without needing to track individual roots.
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