Question - Simplifying Fractions with Common Denominator
Solution:
To simplify the given expression, we first need to identify a common denominator for the fractions. The terms have the following denominators: 27x^2, 12, and 2(3x). The least common denominator (LCD) among these must be a number that each denominator can divide into without a remainder.The prime factorization of the denominators would be:- For 27x^2: 3^3 * x^2- For 12: 2^2 * 3- For 2(3x): 2 * 3 * xThe LCD needs to account for the highest powers of each prime and variable present in all the denominators, which is: 2^2 * 3^3 * x^2. This simplifies to 108x^2.Now that we have identified the LCD as 108x^2, we can rewrite each fraction with this common denominator:- $$ \frac{5}{27x^2} $$ becomes $$ \frac{5 \cdot 4}{108x^2} $$ because 27x^2 times 4 equals 108x^2.- $$ \frac{5}{12} $$ becomes $$ \frac{5 \cdot 9x^2}{108x^2} $$ because 12 times 9x^2 equals 108x^2.- $$ \frac{2}{2(3x)} $$ becomes $$ \frac{2 \cdot 18x}{108x^2} $$ because 2(3x) times 18x equals 108x^2.Rewriting the expression with the common denominator:\[ \frac{5 \cdot 4}{108x^2} + \frac{5 \cdot 9x^2}{108x^2} + \frac{2 \cdot 18x}{108x^2} \]This simplifies to:\[ \frac{20}{108x^2} + \frac{45x^2}{108x^2} + \frac{36x}{108x^2} \]Now, add the fractions together, noting that the x^2 terms in the second fraction will cancel out with the x^2 in the denominator:\[ \frac{20 + 45x^2 + 36x}{108x^2} \]There are no like terms to combine, so this is as simplified as it gets without factoring the numerator, which is not suggested in this context. Also, there seem to be no common factors between the numerator and the denominator that can be cancelled out.So the final simplified expression is:\[ \frac{20 + 45x^2 + 36x}{108x^2} \]
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