Final answer:
Equivalent expressions with the least common denominator for (7)/(3v^6w) and (10)/(27vw^5) are (7w^4)/(3v^6w^5) and (10)/(27v^6w^5), achieved by scaling each fraction appropriately.
Step-by-step explanation:
The goal is to find equivalent expressions that have the least common denominator for the fractions (7)/(3v^6w) and (10)/(27vw^5). First, identify the least common denominator (LCD). The LCD must accommodate both v^6 and v, and w and w^5. Therefore, the LCD is 3v^6w^5 since we need the highest powers of the variables present in both denominators.
Now, we look at how to scale each fraction to have this common denominator:
- For (7)/(3v^6w), multiply the numerator and denominator by w^4 to get (7w^4)/(3v^6w^5).
- For (10)/(27vw^5), multiply the numerator and denominator by 3v^5 to get (30v^5)/(81v^6w^5). Since both denominators must be identical for the fractions to be equivalent, reduce (30v^5)/(81v^6w^5) to (10)/(27v^6w^5) by dividing numerator and denominator by 3v.
The equivalent expressions with the least common denominator are (7w^4)/(3v^6w^5) and (10)/(27v^6w^5).