Problem 42 - Intermediate Algebra

Question

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ b^{9 / 5} b^{8 / 5} $$

Step-by-Step Solution

Answer

The simplified expression is \( b^{17/5} \).

Step 1: Understanding the Problem

We need to simplify the expression \( b^{9/5} \cdot b^{8/5} \). This involves using the properties of exponents to combine these into a single term with positive exponents.

Step 2: Applying Exponent Product Rule

According to the product rule of exponents, when multiplying two exponential expressions that have the same base, you add their exponents. Thus, \( b^{9/5} \cdot b^{8/5} = b^{(9/5 + 8/5)} \).

Step 3: Calculating the Sum of the Exponents

Add the exponents in the expression: \( \frac{9}{5} + \frac{8}{5} = \frac{9 + 8}{5} = \frac{17}{5} \).

Step 4: Writing the Simplified Expression

Now we write the expression with the resulting exponent: \( b^{17/5} \). This expression already uses positive exponents, so we don't need further simplification.

Key Concepts

Positive Exponents

Exponents represent how many times a number, called the base, is multiplied by itself. When we talk about positive exponents, we're referring to exponents that are greater than zero. Positive exponents are straightforward because they signify repeated multiplication without any inverse operations like division.
For example, if you see an expression like \( b^3 \), it simply tells us to multiply the base \( b \) by itself three times: \( b \times b \times b \).
Using positive exponents helps keep expressions simple and avoids complex calculations or interpretations usually associated with negative exponents. Remember:

• Positive means multiplication: \( b^n \) means \( b \times b \times \cdots \times b \) (\( n \) times).
• A positive exponent is a friendly guide that shows repeated multiplication as a shortcut.
• Always aim to express your final answers with positive exponents for clarity and simplicity.

Keeping exponents positive is key when simplifying expressions.

Product Rule of Exponents

The product rule is a fundamental property of exponents that greatly simplifies multiplication of similar bases. This rule states that when multiplying two expressions that have the same base, you can simply add their exponents. This is because multiplying powers of the same base is essentially adding up the number of times you're multiplying the base.
For example: if you have \( b^2 \) multiplied by \( b^3 \), using the product rule gives you \( b^{2+3} = b^5 \).\
In the example exercise, we apply the product rule to \( b^{9/5} \cdot b^{8/5} \). This means adding the exponents: \( \frac{9}{5} + \frac{8}{5} = \frac{17}{5} \).

• Always ensure the bases are identical before applying the product rule.
• Simply add the exponents, which is a quick method to simplify expressions with the same base.

This rule significantly reduces complexity in calculations and makes simplifying expressions easier.

Simplifying Expressions

Simplifying expressions is the process of making them as straightforward as possible. Using properties of exponents, like the product rule, allows us to condense expressions into their simplest forms.
The aim is to deal with fewer multiplication operations by leveraging the power of exponent mathematics.
In the given exercise, we simplified \( b^{9/5} \cdot b^{8/5} \) to \( b^{17/5} \) by using the product rule.
Simplifying is crucial for several reasons:

• It minimizes errors by reducing complex steps and calculations.
• Makes calculations easier and faster to work with, especially for larger expressions.
• Presents information clearly, which is essential in math and science.

By focusing on using positive exponents, and applying appropriate rules like the product rule, we're able to simplify expressions efficiently.