(a/3+3/a+2)*1/a+3

3 min read Jun 16, 2024

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.