Problem 54 - A Graphical Approach to College Algebra

Question

Find each product. Assume that all variables represent positive real numbers. $$p^{11 / 5}\left(3 p^{4 / 5}+9 p^{19 / 5}\right)$$

Step-by-Step Solution

Answer

The product is \(3p^3 + 9p^6\).

Step 1: Distribute the Outer Term

The given expression is \(p^{11/5}(3p^{4/5} + 9p^{19/5})\). To find the product, distribute \(p^{11/5}\) into the terms inside the parentheses. This involves multiplying \(p^{11/5}\) with each term in the parenthesis separately.

Step 2: Multiply First Term

Multiply \(p^{11/5}\) with the first term inside the parenthesis: \[p^{11/5} \cdot 3p^{4/5} = 3 \cdot p^{11/5 + 4/5} = 3p^{15/5} = 3p^3\]This is because when multiplying powers of the same base, you add the exponents.

Step 3: Multiply Second Term

Now, multiply \(p^{11/5}\) with the second term inside the parenthesis:\[p^{11/5} \cdot 9p^{19/5} = 9 \cdot p^{11/5 + 19/5} = 9p^{30/5} = 9p^6\]Adding the exponents \(\frac{11}{5} + \frac{19}{5} = \frac{30}{5} = 6\).

Step 4: Combine the Terms

Finally, combine the results from Step 2 and Step 3:\[3p^3 + 9p^6\]This is the expanded form of the original expression.

Key Concepts

Product of Powers

When dealing with polynomial expressions, it's crucial to understand how to multiply powers of the same base. This is known as the "Product of Powers" property.

• Consider the expression where you multiply two terms like \(p^{11/5}\) and \(p^{4/5}\).
• The base \(p\) remains the same, while the exponents \(\frac{11}{5}\) and \(\frac{4}{5}\) are added together.

Adding these exponents gives \(\frac{11}{5} + \frac{4}{5} = \frac{15}{5} = 3\), which simplifies to \(p^3\).

So, when multiplying powers, the key idea is to keep the base and add the exponents. This concept makes handling polynomial expressions straightforward, especially when they involve products of powers with the same base.

Distributive Property

The distributive property is an essential algebraic property, particularly when handling expressions within parentheses. This property helps us simplify expressions efficiently. Here's how it works:

• Take an expression like \(p^{11/5}(3p^{4/5} + 9p^{19/5})\).
• Using the distributive property, distribute \(p^{11/5}\) to each term in the parentheses.
• First, \(p^{11/5}\) is multiplied by \(3p^{4/5}\).
• Second, \(p^{11/5}\) is multiplied by \(9p^{19/5}\).

By multiplying each term separately, you apply the distributive property to break down complex polynomial expressions into simpler forms that are easier to manage.

This approach is fundamental when dealing with expressions in algebra, ensuring clarity and simplicity in calculations.

Exponent Rules

The rules governing exponents are vital when dealing with algebraic expressions. They provide a set of guidelines for simplifying and manipulating expressions with powers. Here are some key insights:

• Adding exponents: When you multiply terms with the same base, you add the exponents. For example, \(p^{11/5} \times p^{4/5}\) becomes \(p^{(11/5 + 4/5)} = p^3\).
• Simplifying to integers: Sometimes, adding fractional exponents results in whole numbers, like \(\frac{30}{5} = 6\) when you add \(\frac{11}{5} + \frac{19}{5}\).

Understanding these rules allows for correct manipulation of exponents, making it easier to handle expressions with multiple terms and powers.

By mastering exponent rules, you'll deepen your ability to navigate through polynomial expressions confidently.