Question

question

divide: \frac{\frac{1}{a} + \frac{1}{b}}{\frac{a}{b} + \frac{b}{a}}.

provide your answer below:

Explanation:

Find the LCD of the numerator and denominator

Using the Least Common Denominator knowledge point, we identify the denominators in the complex fraction: \(a\) and \(b\). The least common denominator for both the numerator and the denominator is \(ab\).

Simplify the numerator expression

Using the Complex Rational Expressions knowledge point, we combine the terms in the numerator by finding a common denominator:

$$ \frac{1}{a} + \frac{1}{b} = \frac{b}{ab} + \frac{a}{ab} = \frac{a + b}{ab} $$

Simplify the denominator expression

Using the Complex Rational Expressions knowledge point, we combine the terms in the denominator by finding a common denominator:

$$ \frac{a}{b} + \frac{b}{a} = \frac{a^2}{ab} + \frac{b^2}{ab} = \frac{a^2 + b^2}{ab} $$

Divide the simplified numerator by the denominator

Using the Dividing Rational Expressions knowledge point, we rewrite the division of the complex fraction as multiplication by the reciprocal:

$$ \frac{\frac{a + b}{ab}}{\frac{a^2 + b^2}{ab}} = \frac{a + b}{ab} \cdot \frac{ab}{a^2 + b^2} $$

Simplify the final expression

We cancel the common factor \(ab\) from the numerator and the denominator:

$$ \frac{a + b}{\cancel{ab}} \cdot \frac{\cancel{ab}}{a^2 + b^2} = \frac{a + b}{a^2 + b^2} $$

Answer:

Divide: <blank>\(\frac{a + b}{a^2 + b^2}\)</blank>