(x/7+y/3)(x^2/49+y^2/9-xy/21)

2 min read Jun 17, 2024
(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.

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