- Use the power of a product rule \((a \cdot b)^n = a^n \cdot b^n\) to separate the expression.
- Apply the power of a power rule \((a^m)^n = a^{m \cdot n}\) to simplify the exponents of \(x\) and \(y\), obtaining \(x^{24} y^{-27}\).
- Convert the negative exponent using the rule \(y^{-n} = \frac{1}{y^n}\) to arrive at the final form \(\frac{x^{24}}{y^{27}}\).
- Final Answer: \(\boxed{\frac{x^{24}}{y^{27}}}\).
Explanation
- Problem Analysis
The problem asks to simplify the algebraic expression \((x^8 y^{-9})^3\) and express it using only positive exponents. We are given that \(x\) and \(y\) are non-zero, which ensures that the expression is well-defined. Our goal is to apply the properties of exponents to eliminate the negative exponent.
- Applying Exponent Laws
We use the power of a product rule, which states that \((a \cdot b)^n = a^n \cdot b^n\). Applying this to our expression gives:
\[(x^8 y^{-9})^3 = (x^8)^3 \cdot (y^{-9})^3.\]
Next, we use the power of a power rule, which states that \((a^m)^n = a^{m \cdot n}\).
For the base \(x\): \((x^8)^3 = x^{8 \cdot 3} = x^{24}\).
For the base \(y\): \((y^{-9})^3 = y^{-9 \cdot 3} = y^{-27}\).
Combining these, the expression simplifies to \(x^{24} y^{-27}\).
- Final Simplification and Result
Finally, we need to ensure all exponents are positive. We use the negative exponent rule, which states that \(a^{-n} = \frac{1}{a^n}\).
Applying this to the term \(y^{-27}\), we get:
\[y^{-27} = \frac{1}{y^{27}}.\]
Therefore, the simplified expression is:
\[x^{24} \cdot \frac{1}{y^{27}} = \frac{x^{24}}{y^{27}}.\]
Thus, the expression is simplified to \(\boxed{\frac{x^{24}}{y^{27}}}\).
Examples
In fields like computer science and engineering, simplifying expressions is essential for efficient data representation. For example, when calculating memory addresses in a large data structure or scaling dimensions in a design program, complex formulas with negative exponents often appear. By applying exponent rules, you can transform these into a standard, readable form, making it much easier to perform subsequent calculations or implement them in code. This step-by-step simplification ensures that values remain accessible and manageable, just like converting \((x^8 y^{-9})^3\) into the cleaner form \(\frac{x^{24}}{y^{27}}\).