91Ó°ÊÓ

Exponential expressions are a powerful way to represent repeated multiplication of a quantity. In mathematical terms, an exponential expression has a base and an exponent. This is a compact way to show that a number is being multiplied by itself, multiple times. For example, the expression \(c^{12}\) means the base \(c\) is multiplied by itself 12 times. These expressions allow us to handle very large numbers efficiently and are common in algebra, science, and engineering.

Using exponential expressions simplifies the process of working with large exponents and helps manage operations such as multiplication and division of these expressions easily. When exponentials share a common base, as seen in expressions like \(\frac{c^{12}}{c^{5}}\), specific rules can be applied to simplify them.

The base in an exponential expression is the number that is being multiplied. The exponent tells us how many times to use the base in a multiplication. For instance, in \(c^5\), \(c\) is the base and 5 is the exponent, indicating that \(c\) is used as a factor five times.

The base remains constant in expressions that are simplified using rules such as the division rule of exponents. When the bases are the same in both the numerator and the denominator, as seen in our exercise \(\frac{c^{12}}{c^{5}}\), we can perform operations on the exponents while keeping the base unchanged.

• Base: The constant factor being repeatedly multiplied.
• Exponent: The number indicating how many times the base is used in the multiplication.

Subtraction of exponents happens when we divide two exponential expressions with the same base. According to the division rule of exponents, when we divide, we keep the base and subtract the exponent in the denominator from the exponent in the numerator.

In the exercise \(\frac{c^{12}}{c^{5}}\), both expressions have the base \(c\). Thus, we subtract the exponent in the denominator (5) from the exponent in the numerator (12). This subtraction (\(12 - 5\)) gives us 7, resulting in the simplified expression \(c^7\).

This operation simplifies division of similar bases, allowing us to merge power reductions efficiently. It is a fundamental concept in working with powers and indices, making complex calculations more manageable and straightforward.

• To divide: Subtract the exponent of the denominator from the numerator.
• Maintain the base: Only exponents change.