Find the ragne of the expression `27^(cos 2x) 81^(sin 2x)`

Options

A. -5
B. `1/5`
C. `(1)/(243)`
D. `(1)/(27)`

Answer

To find the range of the expression \( y = 27^{\cos 2x} \cdot 81^{\sin 2x} \), we can follow these steps:

### Step 1: Rewrite the expression in terms of base 3
We start by expressing \( 27 \) and \( 81 \) in terms of base \( 3 \):
\[ 27 = 3^3 \quad \text{and} \quad 81 = 3^4 \]
Thus, we can rewrite \( y \):
\[ y = (3^3)^{\cos 2x} \cdot (3^4)^{\sin 2x} \]
Using the property of exponents \( a^{m} \cdot a^{n} = a^{m+n} \):
\[ y = 3^{3 \cos 2x} \cdot 3^{4 \sin 2x} = 3^{3 \cos 2x + 4 \sin 2x} \]

### Step 2: Find the range of the exponent
Let \( z = 3 \cos 2x + 4 \sin 2x \). We need to find the range of \( z \). The expression \( a \cos x + b \sin x \) can be evaluated using the formula:
\[ \text{Range of } (a \cos x + b \sin x) = [-\sqrt{a^2 + b^2}, \sqrt{a^2 + b^2}] \]
In our case, \( a = 3 \) and \( b = 4 \):
\[ \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
Thus, the range of \( z \) is:
\[ [-5, 5] \]

### Step 3: Determine the range of \( y \)
Now that we have the range of \( z \), we can find the range of \( y \):
\[ y = 3^{z} \]
Since \( z \) ranges from \(-5\) to \(5\), we evaluate \( y \) at these endpoints:
- When \( z = -5 \):
\[ y = 3^{-5} = \frac{1}{3^5} = \frac{1}{243} \]
- When \( z = 5 \):
\[ y = 3^{5} = 243 \]

### Step 4: Conclusion on the range of \( y \)
Thus, the range of \( y \) is:
\[ \left[\frac{1}{243}, 243\right] \]

### Final Answer
The range of the expression \( 27^{\cos 2x} \cdot 81^{\sin 2x} \) is:
\[ \left[\frac{1}{243}, 243\right] \]