Simplify (8a-8b)/(b-a)
The given problem requires the simplification of a rational algebraic expression. Simplification in algebra generally involves reducing the expression to its simplest form by performing operations like addition, subtraction, multiplication, division, and factoring. The provided expression is a fraction where the numerator is 8a minus 8b and the denominator is b minus a. The task is to manipulate and simplify this expression to the most reduced form possible, which often involves factoring out common terms or recognizing patterns that allow for cancellation of terms in the numerator and denominator.
$\frac{8 a - 8 b}{b - a}$
Answer
Solution:
Step 1: Extract the common factor from the numerator.
- Step 1.1: Take out the factor of $8$ from $8a$.
$\frac{8(a) - 8b}{b - a}$ - Step 1.2: Take out the factor of $8$ from $-8b$.
$\frac{8(a) + 8(-b)}{b - a}$ - Step 1.3: Combine the factored terms in the numerator.
$\frac{8(a - b)}{b - a}$
Step 2: Simplify by eliminating the common factors in the numerator and denominator.
- Step 2.1: Extract the factor of $-1$ from $a$.
$\frac{8(-1(-a) - b)}{b - a}$ - Step 2.2: Extract the factor of $-1$ from $-b$.
$\frac{8(-1(-a) - (-b))}{b - a}$ - Step 2.3: Factor out $-1$ from the entire numerator.
$\frac{8(-1(-a + b))}{b - a}$ - Step 2.4: Rearrange the terms in the denominator to match the numerator.
$\frac{8(-1(-a + b))}{-a + b}$ - Step 2.5: Cancel out the common factor in the numerator and denominator.
$\frac{8(-1(\cancel{-a + b}))}{\cancel{-a + b}}$ - Step 2.6: Simplify the expression by dividing $8 \cdot (-1)$ by $1$.
$8 \cdot (-1)$
Step 3: Final computation.
- Multiply $8$ by $-1$ to get the simplified result.
$-8$
Knowledge Notes:
The problem involves simplifying a rational expression. The key steps in simplifying such expressions include:
Factoring: This is the process of breaking down expressions into products of simpler expressions. In this case, we factored out an 8 from the numerator.
Simplifying Fractions: When a numerator and denominator share a common factor, they can be simplified by dividing both by that common factor.
Negative Signs: Factoring out a negative sign can help in recognizing common factors. In this problem, we factored out a $-1$ to turn $b - a$ into $-a + b$, which is equivalent to $-(a - b)$.
Cancellation: If a term appears in both the numerator and denominator, it can be canceled out, provided it's not zero.
Multiplication: The final step often involves multiplying the remaining terms after cancellation to find the simplified expression.
In the given problem, the expression $(8a-8b)/(b-a)$ simplifies to $-8$ by factoring out common terms and canceling them. The negative sign is managed by recognizing that $a - b$ is the negative of $b - a$.
