If the equation 3x3 - 16x2 + mx - 6 = 0 has three positive real roots, out of which two roots are reciprocals of each other, the value of m is _______ .

Let's solve this step by step:

Step 1: Given that two roots are reciprocals of each other, let's assume these roots to be 'a' and '1/a'.

Step 2: The third root is positive, let's assume it to be 'b'.

Step 3: According to Vieta's formulas, the sum of the roots of the equation is equal to the ratio of the second coefficient to the first one. So, a + 1/a + b = 16/3.

Step 4: The product of the roots taken two at a time is equal to the ratio of the third coefficient to the first one. So, a*(1/a)b + ab + b*(1/a) = m/3.

Step 5: The product of all roots is equal to the ratio of the last coefficient to the first one with a change of sign. So, a*(1/a)*b = -(-6/3) = 2.

Step 6: From step 3 and step 5, we can see that a*(1/a) = 2/b. So, b = 2/(a*(1/a)) = 2. And a + 1/a + 2 = 16/3. Solving this, we get a = 3 and 1/a = 1/3.

Step 7: Substituting these values in the equation from step 4, we get 32(1/3) + 32 + 2(1/3) = m/3. Solving this, we get m = 18.

So, the value of m is 18.