## Simplifying the Expression: (a/3 + 3/a + 2) * 1/(a+3)

This article will guide you through the steps of simplifying the expression: **(a/3 + 3/a + 2) * 1/(a+3)**. We will use the order of operations (PEMDAS/BODMAS) and combine like terms to achieve a simplified form.

### Step 1: Find a Common Denominator for the Terms within the Parentheses

The first step is to find a common denominator for the terms within the parentheses: **a/3 + 3/a + 2**. The least common denominator is **3a**.

**a/3**: Multiply numerator and denominator by**a**to get**a²/3a****3/a**: Multiply numerator and denominator by**3**to get**9/3a****2**: Multiply numerator and denominator by**3a**to get**6a/3a**

Now the expression becomes: **(a²/3a + 9/3a + 6a/3a) * 1/(a+3)**

### Step 2: Combine the Terms within the Parentheses

Since all the terms now have the same denominator, we can combine the numerators:

**((a² + 9 + 6a) / 3a) * 1/(a+3)**

### Step 3: Simplify the Expression by Multiplying

Now we can multiply the two fractions:

**(a² + 9 + 6a) / (3a * (a+3))**

### Step 4: Factor the Numerator and Simplify

The numerator can be factored into: **(a+3)(a+3)**

The expression now becomes: **(a+3)(a+3) / (3a * (a+3))**

We can cancel out the common factor **(a+3)** in the numerator and denominator, leaving:

**(a+3) / 3a**

### Conclusion

The simplified form of the expression **(a/3 + 3/a + 2) * 1/(a+3)** is **(a+3) / 3a**.

Remember to note any restrictions on the variable 'a'. In this case, **a cannot equal 0 or -3** as these values would make the denominator zero, resulting in an undefined expression.