Question

1. \\(\frac{3x}{6} + \frac{5}{5xy}\\)

Explanation:

Step1: Find the LCD

The denominators are \(6\) and \(5xy\). The least common denominator (LCD) is the product of the highest powers of all prime factors involved. Prime factors of \(6 = 2\times3\), and for \(5xy\) (prime factors \(5, x, y\)). So LCD is \(2\times3\times5\times x\times y= 30xy\).

Step2: Rewrite fractions with LCD

Rewrite \(\frac{3x}{6}\) with denominator \(30xy\): \(\frac{3x}{6}=\frac{3x\times5xy}{6\times5xy}=\frac{15x^{2}y}{30xy}\).
Rewrite \(\frac{5}{5xy}\) with denominator \(30xy\): \(\frac{5}{5xy}=\frac{5\times6}{5xy\times6}=\frac{30}{30xy}\).

Step3: Add the fractions

Now add the two fractions: \(\frac{15x^{2}y}{30xy}+\frac{30}{30xy}=\frac{15x^{2}y + 30}{30xy}\).
We can factor out 15 from the numerator: \(\frac{15(x^{2}y + 2)}{30xy}=\frac{x^{2}y + 2}{2xy}\) (dividing numerator and denominator by 15).

Answer:

\(\frac{x^{2}y + 2}{2xy}\) (or \(\frac{15x^{2}y + 30}{30xy}\) before simplifying further)