Final answer:
To solve the equation 64ˣ = 256, express both numbers as powers of 2, which are 2⁶ and 2⁸, respectively, set the exponents equal to each other (6x = 8), and solve for x, which is 4/3 or approximately 1.3333.
Step-by-step explanation:
To solve the equation 64ˣ = 256 using the like bases property, we should express both sides of the equation with the same base. We know that 64 can be written as 2⁶ since 2 multiplied by itself 6 times equals 64, and 256 can be written as 2⁸, as 2 multiplied by itself 8 times equals 256. The equation can then be rewritten as:
(2⁶)ˣ = 2⁸
Using the property of exponents where we multiply the exponents when raising a power to another power, the left side becomes 2^(6x). This yields the new equation:
2^(6x) = 2⁸
Since we have a base of 2 on both sides, we can set their exponents equal to each other:
6x = 8
To find the value of x, we divide both sides of the equation by 6:
x = 8/6 = 4/3 or approximately 1.3333.