Simplify (x^2-4x)/(7(x-4))
This problem is asking for the simplification of a rational algebraic expression. The task is to reduce the fraction $(x^2-4x)/(7(x-4))$to its simplest form by performing algebraic manipulations, which typically include factoring and canceling out common factors from the numerator and the denominator. The result should be an equivalent expression that is as simplified as possible.
$\frac{x^{2} - 4 x}{7 \left(\right. x - 4 \left.\right)}$
Answer
Solution:
Step 1: Extract the common factor from the numerator
Extract the common factor $x$ from the terms in the numerator $x^{2} - 4x$.
Step 1.1: Factor out $x$ from $x^{2}$
Write the term $x^{2}$ as $x \cdot x$ and then factor out $x$.
$$\frac{x \cdot x - 4x}{7(x - 4)}$$
Step 1.2: Factor out $x$ from $-4x$
Express $-4x$ as $x \cdot (-4)$ and then factor out $x$.
$$\frac{x \cdot x + x \cdot (-4)}{7(x - 4)}$$
Step 1.3: Combine the factored terms
Combine the factored terms in the numerator to show the common factor explicitly.
$$\frac{x(x - 4)}{7(x - 4)}$$
Step 2: Simplify the fraction by canceling out the common factors
Identify and cancel the common factor $(x - 4)$ in the numerator and denominator.
Step 2.1: Cancel the common factor
Cancel out the common factor $(x - 4)$.
$$\frac{x \cancel{(x - 4)}}{7 \cancel{(x - 4)}}$$
Step 2.2: Write the simplified expression
After canceling out the common factor, write the simplified expression.
$$\frac{x}{7}$$
Knowledge Notes:
To simplify a rational expression, we can follow these steps:
Factorization: Break down the numerator and denominator into their factors. This involves finding the common factor or factors that can be divided out of both the numerator and the denominator.
Common Factors: Identify any common factors in both the numerator and the denominator. A common factor is a term that is present in both parts of the fraction.
Cancellation: Cancel out the common factors. This is based on the property that $\frac{a}{a} = 1$ for any non-zero $a$. When we cancel common factors, we are essentially dividing both the numerator and the denominator by those factors, which simplifies the expression.
Rewriting the Expression: After canceling, rewrite what remains of the expression. If all terms in the numerator and denominator are canceled, the result is 1.
In this problem, we factored out $x$ from the numerator and then canceled the common factor $(x - 4)$, which appeared in both the numerator and the denominator. The final simplified expression is $\frac{x}{7}$.
It's important to note that we can only cancel factors; we cannot cancel terms that are added or subtracted. Also, we cannot cancel anything if the expression is not factored completely.
