(x%5E2%2F49%2By%5E2%2F9-xy%2F21).jpeg "(x/7+y/3)(x^2/49+y^2/9-xy/21)")
Simplifying the Expression: (x/7 + y/3)(x^2/49 + y^2/9 - xy/21)
This expression presents an opportunity to practice algebraic simplification and recognize patterns. We can break down the solution into clear steps:
Step 1: Recognize the Pattern
The expression resembles the expansion of a difference of squares pattern. Notice that:
- (x/7 + y/3) looks like the sum of two terms.
- (x^2/49 + y^2/9 - xy/21) can be rewritten as [(x/7)^2 + (y/3)^2 - (x/7)(y/3)]
This suggests that we can potentially apply the difference of squares formula: (a + b)(a - b) = a^2 - b^2.
Step 2: Rearrange the Terms
Let's rearrange the terms in the second part of the expression:
(x/7 + y/3) [(x/7)^2 + (y/3)^2 - (x/7)(y/3)]
Now we have a clear sum and a difference within the brackets.
Step 3: Apply the Difference of Squares Formula
We can now apply the difference of squares formula:
[(x/7)^2 - (y/3)^2]
Step 4: Simplify
Expanding the squares, we get:
(x^2/49 - y^2/9)
Therefore, the simplified expression for (x/7 + y/3)(x^2/49 + y^2/9 - xy/21) is (x^2/49 - y^2/9).
Key Takeaways
- Recognizing patterns like the difference of squares can simplify complex algebraic expressions.
- Rearranging terms can make it easier to identify applicable formulas.
- Careful application of algebraic rules leads to a simplified expression.