91Ó°ÊÓ

When simplifying expressions with exponents, we use a series of rules known as the properties of exponents. These properties make it easier to work with these types of mathematical expressions. Let’s look at two key properties used in the original solution:

1. **Product of Powers Property**: This property states that when you multiply two powers that have the same base, you add the exponents. Mathematically, this is expressed as: \[a^m \times a^n = a^{m+n}\]So in our example, combining the exponents in the denominator: \[k^{2 / 3} \times k^{-1} = k^{(2/3) + (-1)} = k^{-1/3}\]

2. **Quotient of Powers Property**: This property states that when you divide two powers with the same base, you subtract the exponent of the denominator from the exponent of the numerator. Mathematically, it looks like this: \[a^m / a^n = a^{m-n}\]Using it in our context to simplify the entire expression: \[\frac{k^{1 / 3}}{k^{- 1 / 3}} = k^{1 / 3 - (-1 / 3)} = k^{(1/3) + (1/3)} = k^{2/3}\]
Understanding and applying these properties correctly is essential for simplifying exponent expressions.

Simplification of expressions involves reducing them to their simplest form. Using the properties of exponents greatly assists in this task because it allows us to combine and reduce the terms systematically. Here are the steps we followed in the original solution:

1. **Combine Exponents**: We started by applying the product of powers property to combine the terms in the denominator.

• From: \[ k^{2 / 3} \times k^{-1} \]
• To: \[ k^{-1/3} \]

2. **Rewrite the Expression**: We updated the main fraction using our new simplified denominator.
\[ \frac{k^{1 / 3}}{k^{- 1 / 3}} \]
3. **Subtract Exponents**: Finally, we used the quotient of powers property to subtract the exponents:

• From: \[ k^{1 / 3} / k^{- 1 / 3} \]
• To: \[ k^{1 / 3 - (-1 / 3)} = k^{2 / 3} \]

These steps help us break down and simplify complex expressions in a structured manner. By understanding the rules and applying them methodically, we turn what seems complicated into something simpler and more manageable.

In this context, the problem asks us to assume that all variables represent positive real numbers. This assumption is important for several reasons:

1. **Non-zero Base**: Positive real numbers ensure the base of our exponents is non-zero. This avoids undefined expressions when using properties such as \[a^0 = 1\].

2. **Simplification Validity**: The properties of exponents we use rely on positive bases to be mathematically accurate. For example, both positive real numbers and their inverses (like \[ k^{-1}\]) lead to simplified forms, avoiding issues that can arise with zero or negative bases.

3. **General Assumptions**: It's standard in many problems to assume positive real numbers because this avoids complications and makes the calculations more straightforward.

Suppose the variable k is positive. This means that operations like multiplication, division, and raising to a power will always yield results within the realm of real numbers, ensuring our simplified form \[ k^{2/3} \] remains valid and interpretable without any imaginary or complex numbers cropping up.

Always remember the underlying assumption of positive real numbers when simplifying expressions in similar exercises, as it helps ensure correctness and clarity in the results.