Final answer:
To solve the exponent equation 8ˣ⁵ = 64ˣ⁵, express both sides with a common base of 8, equate the exponents, and solve for x to find that x = 3.
Step-by-step explanation:
To solve the equation 8ˣ⁵ = 64ˣ⁵, we can start by recognizing that 64 is a power of 8 since 64 is equal to 8². Using this relationship, we can express the equation with a common base:
8¹ˣ⁵ = (8¹²)ˣ⁵
8¹ˣ⁵ = 8²ˣ⁵
Now, since the bases are the same, we can set the exponents equal to each other:
x + 5 = 2(x + 1)
x + 5 = 2x + 2
Subtract x from both sides:
5 = x + 2
Finally, subtract 2 from both sides to solve for x:
x = 3
Hence, the solution to the equation is x = 3.