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Simplify. Express your answer as a single fraction in simplest form. \newline4p5+25pq\frac{4p}{5} + \frac{2}{5pq}

Full solution

Q. Simplify. Express your answer as a single fraction in simplest form. \newline4p5+25pq\frac{4p}{5} + \frac{2}{5pq}
  1. Identify common denominator: Identify the common denominator for the fractions 4p5\frac{4p}{5} and 25pq\frac{2}{5pq}. Since both denominators already have a common factor of 55, the least common denominator (LCD) is 5pq5pq.
  2. Adjust fractions: Adjust the fractions so that they both have the common denominator of 5pq5pq. The first fraction, 4p5\frac{4p}{5}, needs to be multiplied by qq\frac{q}{q} to have the denominator 5pq5pq. The second fraction, 25pq\frac{2}{5pq}, already has the common denominator.
  3. Multiply numerator and denominator: Multiply the numerator and denominator of the first fraction by qq to get the common denominator. This gives us (4pq)/(5q)(4p\cdot q)/(5\cdot q) which simplifies to 4pq5pq.\frac{4pq}{5pq}.
  4. Combine fractions: Combine the fractions now that they have the same denominator. This gives us (4pq+2)/5pq(4pq + 2)/5pq.
  5. Check for further simplification: Check if the resulting fraction can be simplified further. Since 4pq4pq and 22 do not have any common factors other than 11, and the denominator 5pq5pq cannot be simplified with the numerator, the fraction is already in its simplest form.

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