Problem 29 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Carry out the indicated operation and write your answer using positive exponents only. $$ x^{2 / 5} \cdot x^{-1 / 5} $$

Step-by-Step Solution

Answer

The short answer is: \(x^{\frac{1}{5}}\).

Step 1: Identify the rule of exponents

Since we are given an expression with exponents having the same base, we can apply the rule for multiplying numbers with the same base which states that: \(a^m \cdot a^n = a^{m+n}\) In this case, our base is \(x\) and the two exponents are \(\frac{2}{5}\) and \(-\frac{1}{5}\).

Step 2: Apply the rule of exponents

Using the rule for multiplying numbers with the same base, we can add the exponents to find the final answer: \(x^{\frac{2}{5}} \cdot x^{-\frac{1}{5}} = x^{\frac{2}{5} + (-\frac{1}{5})}\) Now, we just need to simplify the expression inside the exponent.

Step 3: Simplify the expression inside the exponent

Adding the exponents, we get: \(\frac{2}{5} - \frac{1}{5} = \frac{2-1}{5} = \frac{1}{5}\)

Step 4: Write the final answer using positive exponents only

Since we have simplified the expression inside the exponent, we can now write the final answer as: \(x^{\frac{1}{5}}\) The operation has been carried out and the final answer is in the form of a positive exponent.

Key Concepts

Rules of Exponents

Exponentiation involves rules that help simplify expressions with powers. When you multiply two terms that have the same base, you add the exponents. This is because multiplying terms involves repeatedly multiplying the base, so it makes sense to combine the exponents into a single power.
In mathematical terms, for any number or expression with the same base \(a\), the rule is:

• \( a^m \cdot a^n = a^{m+n} \)

This rule helps us work with exponential expressions more efficiently, especially when dealing with more complex expressions. Recognizing the base and correctly identifying the exponents is crucial for applying this rule effectively. In our exercise, the base \(x\) appears in both terms, allowing us to use this rule to simplify the expression.

Simplifying Expressions

Simplifying expressions with exponents is about making them more straightforward to understand and use. Once you've applied a rule like the one for multiplying exponents, you should proceed by simplifying any arithmetic calculations in the powers.
In our case, we had the expression \(x^{\frac{2}{5} + (-\frac{1}{5})}\), which simplifies to \(x^{\frac{2-1}{5}}\). Calculating \(\frac{2-1}{5}\) gives us \(\frac{1}{5}\).
This simplification step is essential. It boils down the expression to its most basic form using simple arithmetic. It also ensures that we can focus on other operations if necessary without dealing with complex or cumbersome expressions. Simplified expressions are easier to interpret and apply in broader mathematical problems.

Positive Exponents

Writing expressions with positive exponents is typically preferred because it's a more standard and accepted form in mathematics, making expressions easier to read and work with. Negative exponents indicate division and can be converted to positive ones by reciprocating the base.
In this exercise, even though the initial expression included a negative exponent, \(x^{-1/5}\), our application of the rules of exponents moved us toward a single expression that features a positive exponent: \(x^{\frac{1}{5}}\).
Converting and simplifying into positive exponents ensures clarity and consistency in mathematical communication. It also aids in preventing errors, especially when transferring calculations into more complex contexts. Aim for positive exponents wherever possible to streamline the simplification process and keep expressions neat and comprehensible.