Final answer:
To solve the equation for b, we rewrite 8 as 2^3, express both sides with the base of 2, and equate the exponents. The equation simplifies to 3b-3 = 4b+4. Solving this linear equation, we find that b = -7.
Step-by-step explanation:
To solve the equation 8^b^-^1 = 2^4^b^+^4, we need to express both sides with the same base and then equate the exponents.
Since 8 is 2^3, we can rewrite 8^b as (2^3)^b. Hence, our equation becomes:
(2^3)^(b-1) = 2^(4b+4)
Simplifying the left side by multiplying the exponents gives:
2^(3b-3) = 2^(4b+4)
Now that both sides have the same base, we can equate the exponents:
3b - 3 = 4b + 4
Subtracting 4b from both sides:
-b - 3 = 4
Adding 3 to both sides:
-b = 7
Finally, multiplying by -1 to solve for b:
b = -7