91Ó°ÊÓ

Factorization is a crucial concept in algebra that involves breaking down numbers or expressions into their prime components or simpler factors. In this exercise, factorization helps us simplify the numerical coefficients by inspecting the numbers in the expression. Here, the expression \(-36\) and \(+9\) both appear in the fraction. To simplify the fraction, we determine the greatest common factor (GCF) of these numbers to make the next steps more manageable.

• The GCF of \-36\ and \+9\ is \+9\.
• When \(-36\) is divided by \(+9\), the result is \(-4\).

Now, with \(-4\) as the simplified numerical coefficient, the task becomes simpler. Factorization helps in reducing complexity, allowing us to focus on simplifying other parts like variables.

Exponents are a way to express repeated multiplication of a number by itself, and they are represented using the notation \(b^n\), where \(b\) is the base and \(n\) is the exponent. In this particular problem, exponents come into play through variables with different powers.In the process of simplification of \(\frac{-36xb^3}{9xb^2}\), there are different powers of \(b\) present:

• The numerator contains \(b^3\).
• The denominator contains \(b^2\).

To simplify, we use the rule of exponents which states that when we divide terms with the same base, we can subtract the exponents: \[b^m \div b^n = b^{m-n}\]Applying this rule here:

• \(b^3 \div b^2 = b^{3-2} = b^1 = b\)

This simplification of the exponent allows us to combine terms, streamlining the expression into its most concise form.

Numerical coefficients refer to the numbers situated in front of variables or variable expressions. They play a significant role in determining the size or magnitude of the expression. In this exercise, understanding how to work with numerical coefficients is key to simplifying the original problem. In the fraction \(\frac{-36xb^3}{9xb^2}\):

• The top (numerator) includes \-36\ as the numerical coefficient.
• The bottom (denominator) holds \+9\ as the numerical coefficient.

By factoring these numbers, as explained in factorization, we divide them to obtain a new coefficient:

• \(-36 \div 9 = -4\)

Thus, the new numerical coefficient of the simplified expression is \-4\. Understanding this aspect is crucial as it not only influences the expression's resulting value but also helps clarify how numbers interact with variables through simplification.