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Simplifying expressions involves reducing a given mathematical expression into its simplest form. The goal here is to rewrite the expression in such a way that it becomes more concise and easier to understand or solve.
This often involves combining like terms and applying mathematical laws such as the laws of exponents. For example, in the expression \(c d^{4} \cdot c d\), identifying and combining like terms such as \(c \cdot c\) and \(d^{4} \cdot d\) makes it easier to handle.
The final step in simplification is ensuring the expression is as reduced as possible, which in this case, results in \(c^2 d^5\). By simplifying expressions, you can often reveal much clearer and more manageable representations of the problem at hand.

The laws of exponents are mathematical rules that describe how to handle expressions involving powers. These rules make it easier to simplify and manipulate expressions that contain exponents.
A few key rules include:

• When multiplying like bases, add the exponents: \(a^m \cdot a^n = a^{m+n}\).
• When dividing like bases, subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\), provided \(a eq 0\).
• Raising a power to a power involves multiplying the exponents: \((a^m)^n = a^{mn}\).

In our original expression, \(c d^{4} \cdot c d\), we applied the multiplication rule by adding the exponents of like bases. This simplified \(c^1 \cdot c^1\) into \(c^{1+1} = c^2\) and \(d^4 \cdot d^1\) into \(d^{4+1} = d^5\). Understanding these laws helps make complex expressions much more manageable.

Like terms are terms in an algebraic expression that have the same variables raised to the same powers. They can be combined by adding or subtracting their coefficients.
For instance, in algebraic expressions involving multiple variables, like \(c\) and \(d\) in our example, \(c\cdot c\) and \(d^4\cdot d\) are like terms because they have the same base variables. When simplifying, one can combine like terms using the laws of exponents to add the powers.
Recognizing and combining like terms is a fundamental aspect of simplifying expressions. It allows for the expression to be reduced to a simpler form, as in our final simplified version \(c^2 d^5\). By understanding this concept, simplifying expressions becomes a straightforward task.