Math Problem Statement

Simplify the expression: (y^2 - 16) / (y^2 + 8y + 16) × (y + 4) / (y^2 - 8y + 16)

Solution

We are given the following expression:

y216y2+8y+16×y+4y28y+16\frac{y^2 - 16}{y^2 + 8y + 16} \times \frac{y + 4}{y^2 - 8y + 16}

Step 1: Factor each term

  1. First numerator: y216y^2 - 16 is a difference of squares, so: y216=(y4)(y+4)y^2 - 16 = (y - 4)(y + 4)

  2. First denominator: y2+8y+16y^2 + 8y + 16 is a perfect square trinomial, so: y2+8y+16=(y+4)2y^2 + 8y + 16 = (y + 4)^2

  3. Second denominator: y28y+16y^2 - 8y + 16 is also a perfect square trinomial, so: y28y+16=(y4)2y^2 - 8y + 16 = (y - 4)^2

Step 2: Rewrite the expression with factored terms

(y4)(y+4)(y+4)2×y+4(y4)2\frac{(y - 4)(y + 4)}{(y + 4)^2} \times \frac{y + 4}{(y - 4)^2}

Step 3: Simplify by canceling common factors

  • The factor (y+4)(y + 4) appears once in the numerator and once in the denominator, so cancel one (y+4)(y + 4).
  • Similarly, the factor (y4)(y - 4) appears in both, so cancel one (y4)(y - 4).

After canceling, we are left with:

1(y+4)(y4)=1y216\frac{1}{(y + 4)(y - 4)} = \frac{1}{y^2 - 16}

Final Answer:

The simplified expression is:

1y216\frac{1}{y^2 - 16}


Would you like more details on any of these steps?

Here are 5 related questions:

  1. How do you factor a difference of squares?
  2. What are perfect square trinomials, and how do you factor them?
  3. Can you always simplify complex rational expressions by canceling factors?
  4. What happens if a variable in the denominator equals zero?
  5. How do you multiply two rational expressions?

Tip: Always check for common factors before multiplying rational expressions, as canceling early can save time.

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Math Problem Analysis

Mathematical Concepts

Algebra
Factoring
Rational Expressions
Simplification

Formulas

Difference of squares: a^2 - b^2 = (a - b)(a + b)
Perfect square trinomials: a^2 + 2ab + b^2 = (a + b)^2, a^2 - 2ab + b^2 = (a - b)^2

Theorems

Factoring Theorem

Suitable Grade Level

Grades 9-11