Question

Simplify: \(\left(x^{-3} y^{5}\right)^{-2}\)

Short answer

\(\frac{x^6}{y^{10}}\)

Step-by-step solution

Apply the Power Rule

Use the power rule of exponents \( (a^m)^n = a^{mn} \) to simplify the expression. Here the expression is \( \left(x^{-3} y^{5}\right)^{-2} \), which becomes \((x^{-3})^{-2} (y^{5})^{-2} = x^{6} y^{-10}\).

Simplify Negative Exponents

Convert the negative exponent of \( y \) to a positive exponent by writing it as a fraction. \( y^{-10} = \frac{1}{y^{10}} \). Since \ x^{6} \ already has a positive exponent, it remains in the numerator. The expression now becomes \ \frac{x^6}{y^{10}} \.

Key concepts

Understanding the Power Rule

The power rule is a fundamental concept in algebra that simplifies expressions involving exponents. It states that when you raise a power to another power, you multiply the exponents. This can be written as \((a^m)^n = a^{mn}\).

For example, let's simplify the expression \((x^{-3} y^5)^{-2}\).

• First, apply the power rule to each term: \((x^{-3})^{-2}\) and \((y^5)^{-2}\).
• Using the rule, it becomes \((x^{-3})^{-2} = x^{(-3) \times (-2)} = x^6\).
• Similarly, \((y^5)^{-2} = y^{5 \times (-2)} = y^{-10}\).

So, \((x^{-3}y^5)^{-2} = x^6 y^{-10}\).

Working with Negative Exponents

Negative exponents can be confusing at first, but they're easy to handle once you understand the rules. A negative exponent indicates division, not multiplication. Specifically, \(a^{-m} = \frac{1}{a^m}\).

In our example, after applying the power rule, we got \x^6 y^{-10}\. The \x^6\ remains in the numerator because it has a positive exponent.

To convert \y^{-10}\ to a positive exponent:

• Use the rule for negative exponents: \y^{-10} = \frac{1}{y^{10}}\.

This changes the expression to:
\[ \frac{x^6}{y^{10}} \]
which is now simplified with all positive exponents.

Simplifying Expressions with Exponents

Simplifying expressions with exponents involves several steps and rules. Key things to remember include:

• **Power Rule:** When raising a power to another power, multiply the exponents like \( (a^m)^n = a^{mn} \).
• **Negative Exponents:** To convert negative exponents to positive, use division \( a^{-m} = \frac{1}{a^m} \).
• **Basics:** Ensure each term has positive exponents for the simplest form.

Let's review our example: \ (x^{-3} y^{5})^{-2} = \frac{x^6}{y^{10}} \.

As seen above, applying these steps systematically simplifies even complex expressions. Always handle one piece at a time to avoid confusion while ensuring every part is in its simplest form.