8. $$ \frac{2 a+1}{a}+\frac{a+4}{4} $$

Question

UpStudy AI · Accepted Answer

To simplify the expression:

$$
\frac{2a + 1}{a} + \frac{a + 4}{4}
$$

**Step 1: Break Down Each Fraction**

First, let's separate each fraction into simpler parts:

$$
\frac{2a + 1}{a} = \frac{2a}{a} + \frac{1}{a} = 2 + \frac{1}{a}
$$
$$
\frac{a + 4}{4} = \frac{a}{4} + \frac{4}{4} = \frac{a}{4} + 1
$$

**Step 2: Combine the Simplified Parts**

Now, add the simplified parts together:

$$
2 + \frac{1}{a} + \frac{a}{4} + 1 = 3 + \frac{a}{4} + \frac{1}{a}
$$

**Step 3: Express as a Single Fraction**

To combine these into a single fraction, find a common denominator, which is $$4a$$:

$$
3 + \frac{a}{4} + \frac{1}{a} = \frac{12a}{4a} + \frac{a^2}{4a} + \frac{4}{4a} = \frac{a^2 + 12a + 4}{4a}
$$

**Final Simplified Form:**

$$
\frac{a^2 + 12a + 4}{4a}
$$

This is the simplest form of the given expression.
The simplified form of the expression is $$ \frac{a^2 + 12a + 4}{4a} $$.

8. \( \frac{2 a+1}{a}+\frac{a+4}{4} \)