Questions: Use a combination of rules for exponents to simplify the expression. Give your answer in exponential form, using only positive exponents. k (x^(-2) y/z^(4))^(-3) (Simplify your answer. Type exponential notation using positive exponents.)

Solution Steps

To simplify the given expression using the rules for exponents, follow these steps:

1. Apply the power of a quotient rule: \((\frac{a}{b})^n = \frac{a^n}{b^n}\).
2. Apply the power of a power rule: \((a^m)^n = a^{m \cdot n}\).
3. Simplify the expression by converting negative exponents to positive exponents using the rule: \(a^{-n} = \frac{1}{a^n}\).

We start with the expression \( k \left( \frac{x^{-2} y}{z^{4}} \right)^{-3} \). By applying the power of a quotient rule, we rewrite it as: \[ k \cdot \frac{(x^{-2} y)^{-3}}{(z^{4})^{-3}} \]

Next, we apply the power of a power rule to each part of the expression: \[ k \cdot \frac{x^{6} y^{-3}}{z^{-12}} \]

Now, we convert the negative exponents to positive exponents: \[ k \cdot \frac{x^{6}}{y^{3}} \cdot z^{12} \]

Combining the terms, we arrive at the simplified expression: \[ \frac{x^{6} z^{12}}{y^{3}} \]

Final Answer