Simplify (8/(p-1))÷(8/(2p^2-2p))

Solution:

Step 1:

Convert the division of fractions into multiplication by the reciprocal. $\frac{8}{p-1}\times \frac{2p^2-2p}{8}$

Step 2:

Eliminate the common factor, which is $8$.

Step 2.1:

Remove the common factor. $\frac{\cancel{8}}{p-1}\times \frac{2p^2-2p}{\cancel{8}}$

Step 2.2:

Reformulate the expression. $\frac{1}{p-1}(2p^2-2p)$

Step 3:

Distribute $\frac{1}{p-1}$ across $2p^2-2p$. $\frac{2p^2-2p}{p-1}$

Step 4:

Extract the common term, $2p$, from the numerator.

Step 4.1:

Extract $2p$ from $2p^2$. $\frac{2p(p)-2p}{p-1}$

Step 4.2:

Extract $2p$ from $-2p$. $\frac{2p(p)+2p(-1)}{p-1}$

Step 4.3:

Factor out $2p$ from the entire numerator. $\frac{2p(p-1)}{p-1}$

Step 5:

Cancel out the common term, $p-1$.

Step 5.1:

Remove the common term. $\frac{2p(\cancel{p-1})}{\cancel{p-1}}$

Step 5.2:

Simplify to get the final result. $2p$

Knowledge Notes:

To solve the given problem, we apply the following mathematical concepts and rules:

1. Division of Fractions: To divide by a fraction, you multiply by its reciprocal. This is based on the rule that $\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}\times \frac{d}{c}$.

2. Simplifying Fractions: When you have common factors in the numerator and the denominator, you can cancel them out to simplify the fraction.

3. Factoring: This involves taking out a common factor from terms in an expression. For example, in the expression $2p^2-2p$, the common factor is $2p$, and factoring it out gives us $2p(p-1)$.

4. Cancellation: If a term appears in both the numerator and the denominator of a fraction, it can be cancelled out, provided it is not zero. This is because any number divided by itself is equal to one.

By applying these rules systematically, we can simplify the given expression step by step to reach the final simplified form.