Simplify (p+(2p)(5))/6

Solution:

Step 1: Simplify the expression in the numerator.

• Step 1.1: Extract the common factor $p$ from the terms $p + 2p \cdot 5$.

  • Step 1.1.1: Express $p$ as $p^1$ to facilitate factoring. $\frac{p^1 + 2p \cdot 5}{6}$
  • Step 1.1.2: Take $p$ out of $p^1$. $\frac{p \cdot 1 + 2p \cdot 5}{6}$
  • Step 1.1.3: Take $p$ out of $2p \cdot 5$. $\frac{p \cdot 1 + p(2 \cdot 5)}{6}$
  • Step 1.1.4: Combine the factored $p$ from both terms. $\frac{p(1 + 2 \cdot 5)}{6}$

• Step 1.2: Perform the multiplication inside the parentheses. $\frac{p(1 + 10)}{6}$

• Step 1.3: Sum the numbers within the parentheses. $\frac{p \cdot 11}{6}$

Step 2: Rearrange the terms in the numerator.

• Step 2: Position the constant before the variable. $\frac{11p}{6}$

Knowledge Notes:

The problem requires simplifying a rational expression. The process involves several algebraic techniques:

1. Factoring: This is the process of breaking down an expression into its constituent factors. In this case, we factor out the common variable $p$ from the terms in the numerator.

2. Combining like terms: After factoring, we combine terms that are similar, which in this case are the terms that involve the variable $p$.

3. Simplifying expressions: This involves performing arithmetic operations such as multiplication and addition within the parentheses to simplify the expression further.

4. Rearranging terms: In algebra, it is conventional to write constants before variables when they are multiplied together. This is more of a stylistic choice than a mathematical necessity, but it makes the expression neater and often easier to read.

The expression is simplified by first factoring out the common variable, then performing the arithmetic operations, and finally rearranging the terms to achieve the simplest form. The final result is a simplified rational expression.