Certainly! Let's simplify the given expression step-by-step.
### Given Expression:
[tex]\[ 2(a - b) + 4\left(\frac{1}{2a} a + 6b\right) \][/tex]
### Step 1: Simplify the expression inside the parentheses
First, let's look at the term inside parentheses:
[tex]\[ \frac{1}{2a} a + 6b \][/tex]
Notice that [tex]\(\frac{1}{2a} a\)[/tex] simplifies directly because the [tex]\(a\)[/tex] in the numerator and denominator cancel out:
[tex]\[ \frac{1}{2a} a = \frac{a}{2a} = \frac{1}{2} \][/tex]
So, now we have:
[tex]\[ \frac{1}{2} + 6b \][/tex]
### Step 2: Distribute the 4 into the expression inside the parentheses
Next, distribute the 4:
[tex]\[ 4 \left( \frac{1}{2} + 6b \right) = 4 \cdot \frac{1}{2} + 4 \cdot 6b \][/tex]
[tex]\[ = 2 + 24b \][/tex]
### Step 3: Combine with the first part of the expression
Bringing everything together, we now have:
[tex]\[ 2(a - b) + 2 + 24b \][/tex]
### Step 4: Distribute the 2 into the first part of the expression
Next, distribute the 2:
[tex]\[ 2(a - b) = 2a - 2b \][/tex]
So now we combine everything:
[tex]\[ 2a - 2b + 2 + 24b \][/tex]
### Step 5: Combine like terms
Now, let's combine the like terms:
- Combine the [tex]\( -2b \)[/tex] and [tex]\( 24b \)[/tex] terms:
[tex]\[ -2b + 24b = 22b \][/tex]
- We add the constant term 2.
Thus, the simplified form is:
[tex]\[ 2a + 22b + 2 \][/tex]
So the final simplified expression is:
[tex]\[ 2a + 22b + 2 \][/tex]
