Math Problem Statement

Simplify the expression: (2p / (p^2 - q^2)) ÷ (4q / (p + q)).

Solution

The expression in the image appears to be:

2pp2q2p+q4q.\frac{2p}{p^2 - q^2} \cdot \frac{p + q}{4q}.

Let's simplify this step by step:

Step 1: Factorize the denominator p2q2p^2 - q^2

The denominator p2q2p^2 - q^2 is a difference of squares, which can be written as: p2q2=(pq)(p+q).p^2 - q^2 = (p - q)(p + q).

So the expression becomes: 2p(pq)(p+q)p+q4q.\frac{2p}{(p - q)(p + q)} \cdot \frac{p + q}{4q}.


Step 2: Simplify the multiplication

Now multiply the numerators and denominators: 2p(p+q)(pq)(p+q)4q.\frac{2p \cdot (p + q)}{(p - q)(p + q) \cdot 4q}.

Cancel out the common factor p+qp + q (assuming p+q0p + q \neq 0): 2p(pq)4q.\frac{2p}{(p - q) \cdot 4q}.


Step 3: Simplify further

Combine 2p2p and 4q4q in the denominator: 2p4q(pq)=p2q(pq).\frac{2p}{4q(p - q)} = \frac{p}{2q(p - q)}.


Final Simplified Expression:

p2q(pq).\frac{p}{2q(p - q)}.

Would you like me to provide additional details or further explain any steps?


Related Questions:

  1. How do you factorize other types of polynomials like a2+2ab+b2a^2 + 2ab + b^2?
  2. Why is the difference of squares formula p2q2=(pq)(p+q)p^2 - q^2 = (p - q)(p + q) important in algebra?
  3. What conditions must hold true for terms to be canceled in fractions?
  4. How do you handle cases where pq=0p - q = 0 or p+q=0p + q = 0?
  5. What are some practical applications of simplifying algebraic fractions?

Tip:

Always check for factors that can be canceled, but ensure that the terms you're canceling are not equal to zero!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Algebra
Factoring
Simplifying Rational Expressions

Formulas

Difference of squares: a^2 - b^2 = (a - b)(a + b)

Theorems

Properties of Rational Expressions
Multiplication and Division of Fractions

Suitable Grade Level

Grades 8-10