Problem 79 - Elementary and Intermediate Algebra

Question

Simplify each expression. Write the answers without negative exponents. All variables represent positive real numbers. See Example 8. $$ 6^{-2 / 3} 6^{-4 / 3} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{1}{36} \).

Step 1: Apply Negative Exponent Rule

To simplify the expression \( 6^{-2/3} \times 6^{-4/3} \), start by applying the rule for multiplying powers with the same base, which is to add the exponents. The expression becomes \( 6^{(-2/3) + (-4/3)} \).

Step 2: Add Exponents

Add the exponents: \(-2/3 + (-4/3) = -6/3 = -2\). This transforms our expression to \( 6^{-2} \).

Step 3: Convert to Positive Exponent

Convert the expression \( 6^{-2} \) to have a positive exponent by using the property \( a^{-b} = \frac{1}{a^b} \). Thus, \( 6^{-2} = \frac{1}{6^2} \).

Step 4: Simplify the Expression

Calculate \( 6^2 \): \( 6^2 = 36 \). Therefore, \( \frac{1}{6^2} \) simplifies to \( \frac{1}{36} \).

Key Concepts

Understanding Positive Real Numbers

In mathematics, a real number is a value that represents a quantity along a continuous line. Real numbers include both rational numbers (like 5, 8.1, or -3.33) and irrational numbers (such as \( \sqrt{2} \) or \( \pi \)). Positive real numbers are simply those real numbers that are greater than zero. Imagine a number line starting at zero and stretching infinitely to the right. Positive real numbers are all the numbers that fall to the right of zero on this line.

In the context of simplifying expressions, it's important to note that positive real numbers have certain predictable behaviors when used in various mathematical operations. For instance:

• When multiplied by another positive real number, the result is always positive.
• Positive numbers are also straightforward to manage when considering powers and roots, which is why they are often used in exercises involving exponents.

By understanding and recognizing positive real numbers, you prepare yourself to navigate and simplify mathematical expressions more effectively.

How to Simplify Expressions

Simplifying expressions involves reducing them to their most basic form without changing their values. The goal is to make expressions easier to understand and work with. One common scenario in algebra is dealing with exponents and specifically, negative exponents. Here’s a useful approach:

• Apply the negative exponent rule: Remember that a negative exponent means the division, or reciprocal, of the base raised to the corresponding positive exponent. For example, \( a^{-n} = \frac{1}{a^n} \).
• Add exponents when bases are the same: If you’re multiplying terms with the same base, you can add their exponents together. For instance, \( a^m \times a^n = a^{m+n} \) can help in reducing the expression.

By following these guidelines, even complex expressions with negative exponents can be simplified into a form that’s much easier to interpret.

Multiplying Powers with the Same Base

When you multiply powers that have the same base, the process is straightforward and revolves around a key principle: add the exponents. This is true regardless of whether the exponents are positive, negative, or fractions. Let’s break it down:

Consider the expression \( a^m \times a^n \). Here, \( a \) is the base common to both terms. The overall expression can be simplified by adding \( m \) and \( n \): \( a^m \times a^n = a^{m+n} \).

• This rule greatly simplifies calculations and is fundamental in working with powers in algebra and beyond.
• Even if the exponents are negative, as seen in our original problem, the principle holds. Add the negative exponents to continue simplifying the expression.
• This technique not only simplifies calculations but also helps in converting complex expressions into more manageable ones quickly.

Understanding this process is crucial when simplifying expressions that involve powers, especially when you're tasked with removing negative exponents or simplifying terms for clearer results.