Problem 45 - Intermediate Algebra

Question

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ 3^{1 / 4} \cdot 3^{3 / 8} $$

Step-by-Step Solution

Answer

The simplified expression is \( 3^{\frac{5}{8}} \).

Step 1: Identify the Base and Exponents

Both terms share the same base of 3. The exponents are \( \frac{1}{4} \) and \( \frac{3}{8} \).

Step 2: Use the Product of Powers Property

The product of powers property states that when you multiply powers with the same base, you add their exponents. Therefore, add \( \frac{1}{4} \) and \( \frac{3}{8} \).

Step 3: Find a Common Denominator for the Exponents

To add \( \frac{1}{4} \) and \( \frac{3}{8} \), first convert \( \frac{1}{4} \) to an equivalent fraction with a denominator of 8. Thus, \( \frac{1}{4} = \frac{2}{8} \).

Step 4: Add the Exponents

Now add \( \frac{2}{8} + \frac{3}{8} = \frac{5}{8} \).

Step 5: Write the Simplified Expression

Combine the base and the new exponent to get the simplified expression. The expression is \( 3^{\frac{5}{8}} \).

Key Concepts

Simplifying Expressions

When it comes to simplifying expressions that involve exponents, the goal is to rewrite the expression in its most compact form. This often involves applying the rules of exponents, such as the product of powers property, which allows us to combine like terms effectively.

In our given problem, we have two expressions, each with the same base of 3: \( 3^{\frac{1}{4}} \) and \( 3^{\frac{3}{8}} \). These need to be combined to create a simplified version.

Using exponential properties, particularly when the bases are the same, simplifies the process significantly. This involves straightforward addition of the exponents while keeping the base constant. In this context, it contributes to simplifying complex problems into simpler ones.

Therefore, always look for opportunities to use properties of exponents when you encounter similar bases. This will make the problem-solving process smoother and more intuitive.

Product of Powers Property

The product of powers property is a fundamental principle used in dealing with exponents. It states that when you multiply two exponents with the same base, you simply add their exponents together. This is represented by the formula: \( a^m \cdot a^n = a^{m+n} \).

In our example, the base 3 is the same, so we can apply this property to combine the expressions \( 3^{\frac{1}{4}} \) and \( 3^{\frac{3}{8}} \). To do this, we add the exponents \( \frac{1}{4} \) and \( \frac{3}{8} \).

Before adding the exponents, it's necessary to find a common denominator. In this scenario, the denominators are 4 and 8, and the least common denominator is 8. Therefore, \( \frac{1}{4} \) is converted to \( \frac{2}{8} \), allowing us to carry out a straightforward addition: \( \frac{2}{8} + \frac{3}{8} = \frac{5}{8} \).

This added exponent, \( \frac{5}{8} \), is then used with the base to give the simplified expression: \( 3^{\frac{5}{8}} \). This showcases the efficiency of the product of powers property in solving exponential expressions.

Positive Exponents

Positive exponents are easier to handle when simplifying expressions, as they signify straightforward multiplication. A positive exponent indicates that the base number is multiplied by itself a certain number of times indicated by the exponent.

For instance, \( 3^2 \) refers to multiplying 3 by itself, resulting in 9. The simplifying task, therefore, often involves rewriting expressions so they do not have negative exponents, which require the use of reciprocal operations.

Our problem focuses on maintaining the exponents as positive values throughout the simplification process. It's crucial to add fractions correctly and ensure the final exponent remains positive.

In simplified terms, positive exponents allow for direct calculation and expression, contributing to clearer, concise mathematical results without the need for additional conversions or computations.