Final answer:
To solve the equation 4096^(2x+8) = 262144^(4x+1), we simplify both sides and equate the bases, then solve for x.
Step-by-step explanation:
To solve this equation, we need to simplify both sides and see if they are equal.
Starting with the left side: 40962x+8
Since 4096 is equal to 212, we can rewrite the left side as (212)2x+8. Using the power of exponent rule, we can multiply the exponents: 212(2x+8). This simplifies to 224x+96.
Now, let's simplify the right side: 2621444x+1
Since 262144 is equal to 218, we can rewrite the right side as (218)4x+1. Using the power of exponent rule, we can multiply the exponents: 218(4x+1). This simplifies to 272x+18.
Therefore, the equation is 224x+96 = 272x+18. In order for two exponents to be equal, the bases must be equal. So we equate the bases: 24x+96 = 72x+18. Now we can solve for x:
48x = 78
x = 1.625