Final answer:
To solve the equation, first express all terms with like bases of 2, then apply the laws of exponents to simplify. After equating exponents, solve for the variable 'n'. The final answer is n = 1/3.
Step-by-step explanation:
To solve the exponential equation (1/64)³n × 8 = 2⁹, we need to express everything in terms of like bases. First, recall that 64 = 2⁶ and thus 1/64 = 2⁻⁶. Also, note that 8 = 2³.
Now, we rewrite the equation with these bases:
(2⁻⁶)³n × 2³ = 2⁹
Use the law of exponents which states that when we have a power raised to another power, we multiply the exponents. Therefore, we have:
(2⁻⁶ × 3n) × 2³ = 2⁹
This simplifies to 2⁻18n × 2³ = 2⁹.
To combine the two exponential terms on the left-hand side that share the same base, we add the exponents. So, we get 2⁻18n + ³ = 2⁹, which can be written as 2⁻18n + 3 = 2⁹.
Since the bases are the same, the exponents must be equal, therefore:
⁻18n + 3 = 9
Solving for n, we have:
⁻18n = 6
n = ⁻6/⁻18 = 1/3