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Simplify. Express your answer as a single fraction in simplest form. \newline3w5−23v3w\frac{3}{w^5} - \frac{2}{3v^3w}

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Full solution

Q. Simplify. Express your answer as a single fraction in simplest form. \newline3w5−23v3w\frac{3}{w^5} - \frac{2}{3v^3w}
  1. Identify LCD: Identify the least common denominator (LCD) for the two fractions.\newlineThe LCD for the fractions 3w5\frac{3}{w^5} and 23v3w\frac{2}{3v^3w} is 3v3w53v^3w^5 because it is the smallest expression that both denominators can divide into without leaving a remainder.
  2. Rewrite with common denominator: Rewrite each fraction with the common denominator.\newlineFor the first fraction, multiply both the numerator and the denominator by 3v3w5w5\frac{3v^3w^5}{w^5}, which simplifies to 3v33v^3.\newlineFor the second fraction, multiply both the numerator and the denominator by 3v3w53v3w\frac{3v^3w^5}{3v^3w}, which simplifies to w4w^4.\newlineThe new fractions are 3×3v33v3w5\frac{3 \times 3v^3}{3v^3w^5} and 2×w43v3w5\frac{2 \times w^4}{3v^3w^5}.
  3. Perform multiplications: Perform the multiplications for each fraction.\newlineThe first fraction becomes (9v3)/(3v3w5)(9v^3)/(3v^3w^5).\newlineThe second fraction becomes (2w4)/(3v3w5)(2w^4)/(3v^3w^5).
  4. Combine fractions: Combine the fractions over the common denominator.\newlineNow that both fractions have the same denominator, we can combine them into a single fraction:\newline(9v3)/(3v3w5)−(2w4)/(3v3w5)=(9v3−2w4)/(3v3w5)(9v^3)/(3v^3w^5) - (2w^4)/(3v^3w^5) = (9v^3 - 2w^4)/(3v^3w^5).
  5. Simplify expression: Simplify the expression if possible.\newlineIn this case, there are no common factors in the numerator that can be canceled with the denominator. Therefore, the expression is already in its simplest form.

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