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Negative exponents can be a bit tricky at first, but with practice, they become easy to handle. A negative exponent indicates that the base should be taken as a reciprocal. For example, in the expression \(\frac{1}{a^{n}}\), the base \(a\) to the power of \(n\) when \(n\) is positive can also be written as \(a^{-n}\). So when you encounter a term like \(x^{-6}\), you can rewrite it as \(\frac{1}{x^{6}}\). This transformation is crucial in simplifying rational expressions, especially when consolidating variables in the numerator and denominator.

Consider the term \(x^{-30}\) in our exercise. This can be represented as \(\frac{1}{x^{30}}\). By using this rule, we converted a somewhat complicated-looking negative exponent into a more understandable form—a simple reciprocal.

Understanding the properties of exponents is vital to simplifying algebraic expressions. Here are some of the key properties:

• Product of Powers: \(a^{m} \times a^{n} = a^{m+n}\) - You add the exponents.
• Quotient of Powers: \(a^{m} / a^{n} = a^{m-n}\) - You subtract the exponents.
• Power of a Power: \( (a^{m})^{n} = a^{m \times n}\) - You multiply the exponents.
• Negative Exponent: \( a^{-m} = \frac{1}{a^{m}}\) - You take the reciprocal.

In the exercise, the step where we simplify \(\frac{x^{4}}{x^{-6}} = x^{4-(-6)} = x^{10}\) uses the quotient of powers property. By subtracting the negative exponent from the positive one, we simplified the term effectively.

Simplifying fractions is a fundamental skill in algebra. It involves reducing the fraction to its lowest terms. Here's a basic process:

• Divide Coefficients: Simplify the numeric part as you would any fraction. For instance, \(\frac{-3}{15} \) can be simplified to \( \frac{-1}{5} \).
• Combine Variables: Using the properties of exponents, combine the variables in the numerator and the denominator. For example, \(\frac{x^{4}}{x^{-6}} \) becomes \( x^{10} \).

In our exercise, we first simplify the fraction \(\frac{-3 x^{4} y^{6}}{15 x^{-6} y^{7}}\). We divide the coefficients and we combine similar variables using the properties of exponents. This method ensures that the fraction is in its simplest form before applying any further operations.