Problem 16 - Elementary Algebra

Question

For the following problems, reduce each rational expression to lowest terms. $$ \frac{16 a^{2} b^{3}}{2 a b^{2}} $$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression is \(8b\).

Step 1: Identify factors in the numerator and denominator

First, let's write down the factors of the expression in both the numerator and the denominator. Numerator factors: \(16a^2b^3 = 2^4 * a * a * b * b * b\) Denominator factors: \(2ab^2 = 2*a*b*b\)

Step 2: Cancel common factors

Now that we have the factors written down, we can cancel the common ones (shared by both the numerator and the denominator). In this case, we can cancel the factor of \(2\), one factor of \(a\), and two factors of \(b\). After cancelling: Numerator: \(2^3 * b = 8b\) Denominator: \(1\)

Step 3: Write down the simplified expression

Finally, let's write down the simplified expression after cancelling the common factors. $$ \frac{8b}{1} $$ Since the denominator is just 1, we don't need to write it. The simplified expression is: $$ 8b $$

Key Concepts

Elementary Algebra

Elementary algebra is the most fundamental branch of algebra, concerned with the use of variables to represent numbers in equations and expressions. It's where students first learn to manipulate expressions and understand algebraic concepts. In the given exercise, we work with a rational expression, which is a fraction that has polynomials in both the numerator and the denominator. Simplifying these expressions is a key skill in algebra that will be applied throughout more advanced topics. Simplification can include combining like terms, factoring, and canceling common factors, which are all foundational techniques in algebra.

Factorization

Factorization is the process of breaking down numbers, polynomials, and algebraic expressions into their essential 'factors,' or components that, when multiplied together, yield the original number or expression. In the context of our exercise, we factorize both the numerator and the denominator to identify common factors. A factor is a part of a multiplication that produces the original term. For instance, when we break down the term \(16a^2b^3\), we find its factors: \(2^4\), \(a\), \(a\), \(b\), \(b\), and \(b\). Factorizing allows us to simplify expressions by canceling out common factors in the numerator and the denominator.

Reducing Fractions

Reducing fractions, also known as simplifying, is the process of creating an equivalent fraction in which the numerator and denominator share no common factors other than 1. To reduce a fraction, we look for the greatest common factor (GCF) that the numerator and denominator have in common. We divide both by this GCF to find the reduced form. In simpler cases, like our example with integers and variables, we might cancel out individual common factors directly without finding the GCF. The reduced form of a fraction is often easier to work with and can simplify further calculations.

Common Factors Cancellation

Common factors cancellation is precisely what we did in our step-by-step solution. It's a vital technique used when simplifying rational expressions, most commonly fractions. By canceling common factors, we are reducing the complexity of the expression without changing its value. It's essential to recognize that only factors can be canceled; terms that are added or subtracted cannot be canceled in this manner. In our exercise, we canceled common factors such as \(2\), \(a\), and \(b^2\), which were present in both the numerator and the denominator. This process leads us to a more straightforward form, which is easier to read and use in further operations.