Short Answer 
To solve the equation (dfrac{1}{64} = 15^{2a}), first rewrite it in terms of base powers by expressing 64 as (2^6) and 16 as (2^4). After simplifying the equation to (2^{-6} = 2^{8a}) and using the same base property, you find that (a = dfrac{-3}{4}).
Step 1: Rewrite the Expression
Start by rewriting the original expression to establish a foundation for your calculations. This involves setting the equation using the known values and a consistent format. For example, you can begin with the expression:
- (dfrac{1}{64} = 15^{2a})
Step 2: Convert to Base Powers
The next step requires converting numbers into their corresponding base powers to simplify the equation further. Recognize that 64 can be expressed as a power of 2, and rewrite the equation accordingly. For clarity, rewrite the expression as follows:
- Replace 64 with (2^6) to get: (dfrac{1}{2^6} = 16^{2a})
- Then, further replace 16 with (2^4): (dfrac{1}{2^6} = (2^4)^{2a})
Step 3: Apply the Same Base Property
In this step, you will simplify and evaluate the powers on both sides of the equation. Here, it’s crucial to maintain consistent bases to effectively utilize the properties of exponents. After the simplification, set the exponents equal to one another:
- Your simplified equation will be (2^{-6} = 2^{8a})
- Applying the rule that if bases are the same, then exponents are equal: (-6 = 8a)
- From this, you can solve for (a) to find (a = dfrac{-3}{4})