Problem 43 - Intermediate Algebra

Question

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ x^{-2 / 5} \cdot x^{7 / 5} $$

Step-by-Step Solution

Answer

The simplified expression is \(x\).

Step 1: Understanding the Problem

We are given the expression \(x^{-2/5} \cdot x^{7/5}\). The goal is to simplify this expression using the properties of exponents and write the result with positive exponents.

Step 2: Applying the Product of Powers Property

The Product of Powers Property states that when multiplying like bases, you add their exponents: \(a^m \cdot a^n = a^{m+n}\). Apply this to the expression: \(x^{-2/5} \cdot x^{7/5} = x^{-2/5 + 7/5}\).

Step 3: Adding the Exponents

Add the exponents \(-2/5\) and \(7/5\). To do this, you find a common denominator, which is already 5. Thus, \(-2/5 + 7/5 = (7 - 2)/5 = 5/5\).

Step 4: Simplifying the Result

The result \(x^{5/5}\) simplifies to \(x^1\) since \(5/5 = 1\). The simplified form is \(x^1\), which can be written as \(x\) with a positive exponent.

Key Concepts

Product of Powers Property

When dealing with exponents, it's important to recognize patterns that help simplify expressions efficiently. One such pattern is the "Product of Powers Property." This property states that when multiplying expressions with the same base, you simply add their exponents.

This means, for any base "a," if you have expressions like \(a^m \cdot a^n\), you can rewrite it as \(a^{m+n}\). This is true as long as the base is the same, regardless of what values "m" and "n" take.

• Helps combine terms into a single expression.
• Reduces the complexity by reducing the number of terms.
• A critical step in simplifying expressions and solving problems with powers.

Simplifying Expressions

Simplifying expressions can sometimes feel like solving a puzzle. By leveraging properties like the Product of Powers, you can streamline such tasks effortlessly. Simplification often entails reducing multiple terms with common bases into a single term by combining their exponents.

In our original problem, we started with \(x^{-2/5} \cdot x^{7/5}\). By applying the Product of Powers, we were able to combine these into \(x^{5/5}\). This now reveals the next step, which involves arithmetic simplification.

• Identify terms with common bases.
• Apply property rules to simplify (e.g., addition of exponents).
• Write the expression compactly.

Simplification is key to making equations more manageable and understanding their fundamental characteristics.

Positive Exponents

When working with exponents, maintaining positive exponents is often a normative preference. Positive exponents are easier to interpret and simplify further research or computation tasks. In our expression, we ended with \(x^{5/5}\), which simplifies to \(x^1\) or simply \(x\).

Positive exponents can be thought of as straightforward multiplicative factors. Unlike negative exponents which imply division or reciprocals, simplifying to positive exponents gives a clearer picture of the growth or decay represented by the expression.

• Easier computation and comparison.
• Readily understandable in many applications.
• Representation of straightforward multiplication.

Remember, anytime you simplify to positive exponents, you enhance interpretability and ease of calculation.