Given:
4a²9b²=97
ab=6

To find:
The value of 2a3b

Solution:
We can use the given equations to find the value of a and b, and then use those values to calculate 2a3b.

Step 1: Find the value of a and b

From the given equations, we can write:

4a²9b²=97

Simplifying this equation, we get:

36a²b²=97/4

Now, using the equation ab=6, we can write:

a=6/b

Substituting this value of a in the above equation, we get:

36(6/b)²b²=97/4

Simplifying this equation, we get:

6²b²=97/4

Solving for b, we get:

b=±(2/3)√97/6

Since ab=6, we can use the positive value of b, which gives:

b=(2/3)√97/6

Substituting this value of b in the equation a=6/b, we get:

a=(3/√97)

Step 2: Calculate 2a3b

Using the values of a and b, we can calculate:

2a3b=2(3/√97)(2/3)√97/6

Simplifying this expression, we get:

2a3b=2

Therefore, the value of 2a3b is 2.

Final Answer:
The value of 2a3b is 2.