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Understanding the laws of exponents is essential when working with algebraic expressions. These laws make it easier to manipulate powers and simplify expressions. One of the primary laws states that when you divide powers with the same base, you subtract the exponent of the denominator from the exponent of the numerator. For example, in the expression \( \frac{b^{5}}{b^{-2}} \), we subtract the exponent in the denominator (-2) from the exponent in the numerator (5):
\[ 5 - (-2) = 5 + 2 = 7 \]
This simplification gives us the new exponent. It's important to remember that a negative exponent indicates division by that base raised to the corresponding positive exponent.
Another useful law is the product of powers, where you add the exponents when multiplying powers with the same base.

Coefficients are the numerical parts of terms in an algebraic expression. Simplifying coefficients involves basic arithmetic. For instance, in the original expression \( \frac{12 b^{5}}{4 b^{-2}} \), we start by simplifying the numerical coefficients (12 and 4).
Divide the numerator by the denominator to get:
\[ \frac{12}{4} = 3 \]
This step alone makes the expression much simpler. Always perform this simplification before addressing the variable part of the expression. Simplifying coefficients early makes subsequent steps more straightforward and reduces the complexity of the terms you're working with.

Once you've simplified coefficients and dealt with the exponents, the next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In our simplified expression from previous steps, we have \( 3b^{7} \).
Even though there's only one term left in this expression, understanding combining like terms is crucial for more complex expressions. This means ensuring all terms with the same variable and exponent are combined together. This not only makes the expression simpler but also easier to understand and work with in future operations.
Combining like terms involves adding or subtracting the coefficients of terms that have the same variables and exponents. This simplification process is one of the cornerstones of algebra and helps achieve the simplest form of an expression.