Problem 65 - Elementary Algebra

Question

For the following problems, perform the indicated operations. $$ (r+3)^{4} \cdot \frac{r+4}{(r+3)^{3}} $$

Step-by-Step Solution

Answer

Answer: The simplified expression is $r^2 + 7r + 12$.

Step 1: Simplify the given expression

First, we will rewrite the given expression to make it easier to work with: $$ (r+3)^{4} \cdot \frac{r+4}{(r+3)^{3}} $$

Step 2: Divide exponents with the same base

Since we are dividing two exponents with the same base, we can subtract the powers. Remember that dividing the exponents means subtracting the powers: $$ \frac{(r+3)^{4}}{(r+3)^{3}} = (r+3)^{4-3} = (r+3)^{1} $$

Step 3: Combine the terms

Now, we will combine the terms in the expression, taking into account the simplified exponent: $$ (r+3)^{1} \cdot (r+4) = (r+3)(r+4) $$

Step 4: Expand the expression

Finally, we will expand the expression by applying the distributive property (also known as FOIL): $$ (r+3)(r+4) = r^2 + 4r + 3r + 12 $$

Step 5: Combine like terms and write the final answer

Now, combine the like terms to get the simplified expression: $$ r^2 + 4r + 3r + 12 = r^2 + 7r + 12 $$ The simplified expression after performing the indicated operations is: $$ r^2 + 7r + 12 $$

Key Concepts

Exponent Rules

Exponent rules are crucial when dealing with algebraic expressions involving powers. They provide a foundation for simplifying expressions and solving equations. One of the key rules is when dividing exponents with the same base, which involves subtracting the powers. For example, consider the expression

• \((r+3)^4 \div (r+3)^3\)

According to the exponent rule, this can be simplified by subtracting the power of the divisor from the power of the dividend:

• \((r+3)^{4-3} = (r+3)^1\)

This results in a simpler expression \((r+3)\). Understanding these exponent rules helps streamline the simplification process, making it easier to focus on the remaining parts of the problem.

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply a single term by each term in a parenthesis. It is often remembered by the acronym FOIL when dealing with binomials. In the case of the expression \((r+3)(r+4)\), the distributive property dictates that:

• Multiply the first terms: \(r \times r = r^2\)
• Multiply the outer terms: \(r \times 4 = 4r\)
• Multiply the inner terms: \(3 \times r = 3r\)
• Multiply the last terms: \(3 \times 4 = 12\)

These four products are then added together to form

• \(r^2 + 4r + 3r + 12\)

Using the distributive property helps in breaking down complex expressions into manageable components, providing a straightforward method for expanding and simplifying expressions.

Polynomial Expansion

Polynomial expansion involves multiplying out expressions to eliminate parentheses, making them simpler to work with. It's an extension of the distributive property applied to binomials. From the problem, the expression after applying the distributive property was

• \(r^2 + 4r + 3r + 12\)

To simplify, we combine like terms:

• Add the terms with \(r\) to get: \(4r + 3r = 7r\)

Thus, the fully expanded polynomial becomes:

• \(r^2 + 7r + 12\)

Polynomial expansion is a technique that allows for clear and organized expression of algebraic results, providing the final simplified version of an expression.