Simplify $$ \frac{a^{7} \times a^{-13}}{a^{-5}} $$

When you encounter two exponential expressions with the same base being multiplied, the Product of Powers Property allows you to simplify the expression with ease. The property states that when you multiply two exponents with the same base, you keep the base and add their exponents. For example, if you have \( a^m \times a^n \), the result is \( a^{m+n} \). This property is immensely useful in simplifying expressions and solving equations involving exponents.

Applying this property is straightforward. In our example from the textbook exercise, when we have \( a^7 \times a^{-13} \), we simply add the exponents: \( 7 + (-13) \), which results in \( a^{-6} \). It is crucial to include negative exponents in the calculation as they play a significant role in the outcome.

Dividing exponential expressions with the same base also has its own rule, known as the Quotient of Powers Property. When you divide exponents with the same base, you subtract the exponent of the denominator from the exponent of the numerator. The formula is \( a^m / a^n = a^{m-n} \). This rule makes division involving exponents a more manageable task.

In our exercise's context, when simplifying \( a^{-6} / a^{-5} \), we subtract the exponent in the denominator from the exponent in the numerator: \((-6) - (-5)\). The subtraction of a negative number is equivalent to addition, \( -6 + 5 \) which simplifies to \( a^{-1} \). Remember that the base remains unchanged during this process.

Negative exponents can sometimes be confusing, but they follow a simple rule. An expression with a negative exponent, such as \( a^{-n} \), is equivalent to the reciprocal of the base raised to the positive exponent: \( \frac{1}{a^n} \). They indicate an inverse relationship and should not be mistaken for negative numbers.

The application of this concept in our exercise shows that \( a^{-1} \) is the same as \( \frac{1}{a^1} \) or simply \( \frac{1}{a} \). This conversion from a negative exponent to a positive one is essential in simplifying expressions and understanding their true value.

Algebraic manipulation entails various techniques used to simplify or rearrange algebraic expressions. These techniques include distributing, factoring, combining like terms, and applying the properties of exponents, as we've seen in the Product and Quotient of Powers Properties. Mastering algebraic manipulations requires practice and a grasp of fundamental rules.

Employing algebraic manipulation allows us to find simplified forms of complex expressions. For instance, in simplifying \( \frac{a^{7} \times a^{-13}}{a^{-5}} \), we used algebraic manipulation to apply the properties of exponents correctly. By understanding and manipulating the expressions mathematically, we achieved a simplified result of \( a^{-1} \). This skill is crucial in not only solving textbook exercises but also in solving real-world problems involving variables and numbers.