Problem 39 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Carry out the indicated operation and write your answer using positive exponents only. $$ 2 p^{3 / 2}\left(2 p^{1 / 2}-p^{-1 / 2}\right) $$

Step-by-Step Solution

Answer

The short answer is: \(4p^2 - 2p\).

Step 1: Distribute the term

Multiply \(2p^{3/2}\) with each term inside the parentheses: $$ 2p^{3/2}(2p^{1/2} - p^{-1/2}) $$ Step 2: Simplify each term

Step 2: Simplify each term

Using the rule of exponents \(a^{m}\cdot a^{n} = a^{m + n}\), we can combine the exponents: $$ (2p^{3/2}\cdot 2p^{1/2}) - (2p^{3/2}\cdot p^{-1/2}) $$ Next, add the exponents: $$ (4p^{3/2+1/2}) - (2p^{3/2-1/2}) $$ Simplify the exponents: $$ (4p^{4/2}) - (2p^{2/2}) $$ $$ 4p^2 - 2p^1 $$ The final answer using positive exponents only is: $$ 4p^2 - 2p $$

Key Concepts

Distributive Property

When solving algebraic expressions that involve multiplication over addition or subtraction within parentheses, we employ a fundamental rule known as the distributive property. This property states that we multiply the term outside the parentheses by each term inside the parentheses. For example, if you have an expression like \(2p^{3/2}(2p^{1/2} - p^{-1/2})\), you'll need to distribute \(2p^{3/2}\) to both \(2p^{1/2}\) and \(-p^{-1/2}\). This action essentially "spreads out" the multiplication to each component inside the parentheses, ensuring that no term is left out. This step lays the groundwork for further simplification.

Simplifying Expressions

Simplifying algebraic expressions involves reducing them to their simplest form. After applying the distributive property, you often need to combine like terms or apply laws of exponents to streamline the expression. For instance, when we distribute \(2p^{3/2}\) with \(2p^{1/2}\) and \(-p^{-1/2}\), we need to use the rule \(a^m \cdot a^n = a^{m+n}\) to add the exponents when the bases are the same. This leads to new terms like \(4p^{4/2}\) and \(-2p^{2/2}\). By combining and simplifying these, such as recognizing that \(4/2 = 2\) and \(2/2 = 1\), we refine the expression to \(4p^2 - 2p\). It's crucial to perform these operations correctly to achieve an accurate and simplified end result.

Positive Exponents

Working with positive exponents is a key mathematical principle. Exponents offer a way to denote repeated multiplication of a number by itself. Positive exponents indicate how many times a number, known as the base, is used as a factor. In expressions, it's standard practice to write expressions with positive exponent terms for clarity and simplicity. For example, converting an expression to positive exponents means adjusting terms like \(p^{-1/2}\) to avoid negative powers. In this way, after distributing and simplifying the given expression \(2p^{3/2}(2p^{1/2} - p^{-1/2})\), we convert it into \(4p^2 - 2p\), where all exponent terms are positive. Positive exponents help in making expressions more interpretable and are widely used in algebraic simplification.