91Ó°ÊÓ

To understand the solution to the given exercise, we first need to grasp some basic concepts of algebra. Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols.
These symbols represent numbers and can be used to express general mathematical relationships.

In this exercise, we are dealing with polynomials, which are algebraic expressions consisting of variables and coefficients. The given expression \((r^{6}+s^{6})(r^{6}-s^{6})\) is a perfect example of a polynomial. By understanding how to manipulate and simplify these expressions, we can solve complex problems more efficiently.

When working with algebraic expressions, it's important to recognize patterns and practice the rules of arithmetic systematically. This exercise used the pattern known as the difference of squares, which allows us to easily simplify the product of two binomials.

The exercise focuses on finding the product of two expressions, specifically \((r^{6} + s^{6})(r^{6} - s^{6})\).
Understanding the process involves recognizing the structure and using algebraic identities.

The difference of squares formula is an essential algebraic identity, expressed as:
\[ a^2 - b^2 = (a + b)(a - b) \]
By recognizing that the given expression \((r^{6} + s^{6})(r^{6} - s^{6})\) fits this pattern, we can simplify it effectively.

In this particular problem, we let:
\[ a = r^6 \]
\[ b = s^6 \]
Substituting these values into the formula gives:
\[ (r^{6})^2 - (s^{6})^2 \]
By transforming and simplifying, this results in the expression \((r^6)^2 - (s^6)^2\).

Simplification is the process of making an algebraic expression easier to work with by reducing it to its simplest form.
In this exercise, simplification involves using the power of a power rule, which states:\[ (x^m)^n = x^{mn} \].

Let's simplify the expression step-by-step:

• We start with \((r^{6})^2 - (s^{6})^2\)
• Applying the power of a power rule, we have:
  \[ (r^{6})^2 = r^{12} \]
  
  \[ (s^{6})^2 = s^{12} \]
  
• This simplifies the expression to:
  \[ r^{12} - s^{12} \]

Steps like these make it easier to handle even more complex expressions in algebra. The final result \(r^{12} - s^{12}\) is much simpler to understand and work with than the initial expanded form.
By practicing these techniques, students can build confidence and improve their algebraic manipulation skills.