The number of real solutions of the equation
`27^(1//x)+12^(1//x)=2.8^(1//x)`, is

Options

A. 1
B. 2
C. 0
D. infinite

Answer

To solve the equation \( 27^{\frac{1}{x}} + 12^{\frac{1}{x}} = 2 \cdot 8^{\frac{1}{x}} \), we can follow these steps:

### Step 1: Rewrite the equation in terms of powers
We can express the bases in terms of their prime factors:
\[ 27 = 3^3, \quad 12 = 3 \cdot 2^2, \quad 8 = 2^3 \]
Thus, we can rewrite the equation as:
\[ (3^3)^{\frac{1}{x}} + (3 \cdot 2^2)^{\frac{1}{x}} = 2 \cdot (2^3)^{\frac{1}{x}} \]
This simplifies to:
\[ 3^{\frac{3}{x}} + (3^{\frac{1}{x}} \cdot 2^{\frac{2}{x}}) = 2 \cdot 2^{\frac{3}{x}} \]

### Step 2: Divide the entire equation by \( 2^{\frac{3}{x}} \)
To simplify further, we divide the entire equation by \( 2^{\frac{3}{x}} \):
\[ \frac{3^{\frac{3}{x}}}{2^{\frac{3}{x}}} + \frac{3^{\frac{1}{x}} \cdot 2^{\frac{2}{x}}}{2^{\frac{3}{x}}} = 2 \]
This results in:
\[ \left(\frac{3}{2}\right)^{\frac{3}{x}} + \left(\frac{3}{2}\right)^{\frac{1}{x}} = 2 \]

### Step 3: Let \( T = \left(\frac{3}{2}\right)^{\frac{1}{x}} \)
Now, we can set:
\[ T = \left(\frac{3}{2}\right)^{\frac{1}{x}} \]
Then, we have:
\[ T^3 + T = 2 \]
Rearranging gives us:
\[ T^3 + T - 2 = 0 \]

### Step 4: Factor the cubic equation
We can factor this equation:
\[ T^3 + T - 2 = (T - 1)(T^2 + T + 2) \]
To find the roots, we can set each factor to zero.

### Step 5: Solve for \( T \)
From the factor \( T - 1 = 0 \), we get:
\[ T = 1 \]
For the quadratic \( T^2 + T + 2 = 0 \), we calculate the discriminant:
\[ D = 1^2 - 4 \cdot 1 \cdot 2 = 1 - 8 = -7 \]
Since the discriminant is negative, this quadratic has no real solutions.

### Step 6: Substitute back to find \( x \)
Now substituting \( T = 1 \):
\[ \left(\frac{3}{2}\right)^{\frac{1}{x}} = 1 \]
This implies:
\[ \frac{1}{x} = 0 \quad \Rightarrow \quad x = \infty \]
However, \( x = \infty \) is not a real solution.

### Conclusion
Since the only solution we found is not a real solution, the number of real solutions to the original equation is:
\[ \text{0} \]