Final answer:
The expression a⁷b⁵ - a³b² - a²b⁶ - a²b² is already in its simplest form. Expanding expressions with exponents involves multiplying the exponents when raising one exponent to the power of another. Cubing exponential expressions requires cubing the digit term and multiplying the existing exponent by three.
Step-by-step explanation:
The expression a⁷b⁵ - a³b² - a²b⁶ - a²b² needs to be simplified. This expression cannot be simplified by combining like terms since none of the terms are alike; they all have different exponents on their variables. Therefore, the expression is already in its simplest form. However, you can factor out common terms if the problem context allows for it.
An important rule when working with exponents is that when you raise an expression to a power, you multiply the exponent of each variable inside the expression by that power, which is represented by the rule (xa)b = xa.b.
When dealing with cubing of exponentials, you would cube the digit term as usual and multiply the exponent by 3. This rule is important when you have expressions such as (53)4 which would result in 512 because you multiply the exponents (3*4).
Understanding how to manipulate and simplify expressions with exponents is a fundamental skill in algebra and helps with more advanced mathematical concepts.