\begin{tabular}{l}\hline monomials. SHOW ALL STEPS! \\ \hline 2. \( \frac{k^{6}}{x^{6}} \)\end{tabular}

Certainly! Let's simplify the monomial \( \frac{k^{6}}{x^{6}} \) step by step. ### **Problem 2:** Simplify the monomial \( \frac{k^{6}}{x^{6}} \). --- ### **Step 1: Understand the Given Monomial** We start with the expression: \[ \frac{k^{6}}{x^{6}} \] This is a fraction where both the numerator and the denominator are variables raised to the same exponent. --- ### **Step 2: Apply the Laws of Exponents** One of the fundamental laws of exponents states: \[ \frac{a^{m}}{b^{m}} = \left( \frac{a}{b} \right)^{m} \] where \( a \) and \( b \) are any real numbers (except \( b \neq 0 \)), and \( m \) is an integer. **Applying this law to our monomial:** \[ \frac{k^{6}}{x^{6}} = \left( \frac{k}{x} \right)^{6} \] This step rewrites the original expression by combining the numerator and the denominator into a single fraction raised to the exponent. --- ### **Step 3: Rewrite Using Negative Exponents (Alternative Form)** Another way to express the original monomial is by utilizing negative exponents. Recall that: \[ \frac{1}{a^{m}} = a^{-m} \] **Applying this to the denominator:** \[ \frac{k^{6}}{x^{6}} = k^{6} \cdot x^{-6} \] Here, the denominator \( x^{6} \) is rewritten as \( x^{-6} \), moving it from the denominator to the numerator with a negative exponent. --- ### **Final Simplified Forms** 1. **Combined Fraction Form:** \[ \left( \frac{k}{x} \right)^{6} \] This form is useful when you want to emphasize the relationship between \( k \) and \( x \) as a single entity raised to a power. 2. **Negative Exponent Form:** \[ k^{6} \cdot x^{-6} \] This form is helpful in scenarios where you might combine like terms or further manipulate the expression using other exponent rules. --- ### **Conclusion** Both simplified forms are correct and can be used interchangeably depending on the context or the subsequent steps required in a problem: - \( \left( \frac{k}{x} \right)^{6} \) - \( k^{6} \cdot x^{-6} \) Choose the form that best fits your needs for further calculations or simplifications.