We have to determine y given that 8^x=2 and 3^(x+y)=81.

8^x=2

=> 2^3^x = 2

=> 3x = 1

=> x = 1/3

3^(x + y) = 81

=> 3^(y + 1/3) = 3^4

=> y + 1/3 = 4

=> y = 11/3

**The value of y = 11/3**

It is given that 8^x=2 and 3^(x+y)=81. The value of y has to be determined.

First take 8^x = 2

8 = 2^3

8^x = (2^3)^x = 2^(3x) = 2

This gives 3x = 1 or x = 1/3

Now 3^(x + y) = 81

3^(x + y) = 3^4

As x = 1/3

3^(1/3 + y) = 3^4

As the base is the same equate the exponent

1/3 + y = 4

y = 11/3

The required value of y = 11/3

We'll write 8^x = 2^3x

We'll re-write 8^x = 2 <=> 2^3x = 2

Since the bases are matching, we'll get:

3x = 1

We'll divide by 3:

x = 1/3

We'll use the product property of exponentials:

3^(x+y)=3^x*3^y

We'll substitute x by 1/3 and we'll write 81 as a power of 3:

3^(1/3 + y) = 3^4

Since the bases are matching, we'll get:

1/3 + y = 4

We'll subtract 1/3, to isolate y to the left side:

y = 4 - 1/3

y = 11/3