%2B1%2F(a%2B2)(a%2B4)%2B1%2F(a%2B4)(a%2B6).jpeg "1/a(a+2)+1/(a+2)(a+4)+1/(a+4)(a+6)")
Simplifying the Expression: 1/a(a+2) + 1/(a+2)(a+4) + 1/(a+4)(a+6)
In this article, we will simplify the given expression:
$\frac{1}{a(a+2)} + \frac{1}{(a+2)(a+4)} + \frac{1}{(a+4)(a+6)}$
Step 1: Find the Common Denominator
The least common multiple (LCM) of the denominators is a(a+2)(a+4)(a+6). We will find the equivalent fractions with this common denominator:
$\frac{1}{a(a+2)} = \frac{(a+4)(a+6)}{(a+2)(a+4)(a+6)}$
$\frac{1}{(a+2)(a+4)} = \frac{a(a+6)}{(a+2)(a+4)(a+6)}$
$\frac{1}{(a+4)(a+6)} = \frac{a(a+2)}{(a+2)(a+4)(a+6)}$
Step 2: Add the Fractions
Now, we can add the fractions:
$\frac{(a+4)(a+6) + a(a+6) + a(a+2)}{(a+2)(a+4)(a+6)}$
Step 3: Simplify the Numerator
Combine like terms:
$(a+4)(a+6) + a(a+6) + a(a+2)$ $= a^2 + 10a + 24 + a^2 + 6a + a^2 + 2a$ $= 3a^2 + 18a + 24$
Step 4: Write the Simplified Expression
The simplified expression is:
$\frac{3a^2 + 18a + 24}{(a+2)(a+4)(a+6)}$
Therefore, we have successfully simplified the given expression.
