The product of real roots of the equation `|x|^(6//5)-26|x|^(3//5)-27 =0` is

Options

A. `-3^(10)`
B. `-3^(12)`
C. `- 3^(12//5)`
D. `-3^(21//5)`

Answer

To solve the equation \(|x|^{\frac{6}{5}} - 26|x|^{\frac{3}{5}} - 27 = 0\) and find the product of its real roots, we can follow these steps:

### Step 1: Substitute \(t\)
Let \(t = |x|^{\frac{3}{5}}\). Then, we can express \(|x|^{\frac{6}{5}}\) in terms of \(t\):
\[ |x|^{\frac{6}{5}} = (|x|^{\frac{3}{5}})^2 = t^2 \]
Thus, the equation becomes:
\[ t^2 - 26t - 27 = 0 \]

### Step 2: Factor the Quadratic Equation
Next, we will factor the quadratic equation:
\[ t^2 - 26t - 27 = 0 \]
To factor this, we look for two numbers that multiply to \(-27\) and add to \(-26\). These numbers are \(-27\) and \(1\):
\[ (t - 27)(t + 1) = 0 \]

### Step 3: Solve for \(t\)
Setting each factor to zero gives us:
\[ t - 27 = 0 \quad \Rightarrow \quad t = 27 \]
\[ t + 1 = 0 \quad \Rightarrow \quad t = -1 \]
Since \(t\) represents \(|x|^{\frac{3}{5}}\), we discard \(t = -1\) because it cannot be negative. Thus, we have:
\[ t = 27 \]

### Step 4: Back Substitute for \(|x|\)
Now we substitute back for \(t\):
\[ |x|^{\frac{3}{5}} = 27 \]
To find \(|x|\), we raise both sides to the power of \(\frac{5}{3}\):
\[ |x| = 27^{\frac{5}{3}} = (3^3)^{\frac{5}{3}} = 3^5 = 243 \]

### Step 5: Determine the Values of \(x\)
Since \(|x| = 243\), we have two possible values for \(x\):
\[ x = 243 \quad \text{or} \quad x = -243 \]

### Step 6: Calculate the Product of the Roots
The product of the real roots \(x_1\) and \(x_2\) is:
\[ x_1 \cdot x_2 = 243 \cdot (-243) = -243^2 = -59049 \]
We can express this in terms of powers of 3:
\[ -59049 = -3^{10} \]

### Final Answer
Thus, the product of the real roots is:
\[ \boxed{-3^{10}} \]