Problem 53 - Elementary Algebra

Question

For the following problems, perform the multiplications and divisions. $$ (a-2 b)^{4} \div \frac{(a-2 b)^{2}}{a+b} $$

Step-by-Step Solution

Answer

Question: Simplify the algebraic expression \((a-2b)^4 \div \frac{(a-2b)^2}{a+b}\). Answer: \((a-2b)^2 \cdot (a+b)\)

Step 1: Break down the given expression into two parts

First, let's separate the expression into the following two parts: 1) \((a-2b)^{4}\) 2) \(\frac{(a-2b)^{2}}{a+b}\) These are the two components that we will be working with individually before combining them back together.

Step 2: Simplify each part

Let's break down each part: 1) To simplify \((a-2b)^4\), we should remember that it is equivalent to \(((a-2b)^2)^2\). Leave it for now; we'll simplify it further in a later step. 2) To simplify \(\frac{(a-2b)^2}{a+b}\), we can't do much since there are no common factors in the numerator and denominator. So let's leave it as it is for now.

Step 3: Combine the parts back

Now that we have simplified both parts of the expression, we can rewrite the given expression as follows: $$ (a-2b)^{4} \div \frac{(a-2b)^{2}}{a+b} = (a-2b)^4 \cdot \frac{a+b}{(a-2b)^2} $$

Step 4: Simplify the expression

Now let's simplify the expression. Notice that there is a common factor of \((a-2b)^2\) in both the numerator and the denominator. Therefore, we can cancel out this factor from the expression: $$ (a-2b)^4 \cdot \frac{a+b}{(a-2b)^2} = \frac{(a-2b)^{4}}{(a-2b)^2} \cdot (a+b) $$ $$ = \frac{((a-2b)^2)^2}{(a-2b)^2} \cdot (a+b) $$ $$ = (a-2b)^2 \cdot (a+b) $$ The simplified expression is \((a-2b)^2 \cdot (a+b)\).

Key Concepts

Algebraic Manipulation

Algebraic manipulation involves rewriting expressions to simplify or solve them. This is essential in problems involving polynomials, exponents, and rational expressions. In our exercise, we started by breaking down the expression into two distinct parts:

• \((a - 2b)^4\)
• \(\frac{(a - 2b)^2}{a+b}\)

Next, each part was handled individually before combining them again. This method allows us to simplify complex expressions step-by-step. Manipulation often involves factoring, distributing, and canceling terms to make expressions more manageable. Here, we leveraged the property of division turning into multiplication by the reciprocal to convert the division into multiplication. This switch enabled easier simplification by canceling out common factors. It's crucial to maintain order and precision when performing these steps to ensure accuracy.

Exponents

Exponents are a way to represent repeated multiplication of a number by itself. In expressions, exponents follow specific properties that help in their manipulation. Key exponent properties used in this exercise are:

• Power of a power: \(((a)^m)^n = a^{m \cdot n}\)
• Multiplication of exponents with the same base: \(a^m \cdot a^n = a^{m+n}\)
• Division of exponents with the same base: \(\frac{a^m}{a^n} = a^{m-n}\)

For \((a-2b)^4\), observing that it could also be written as \(((a-2b)^2)^2\) helps recognize shared factors for simplification. After multiplication is expressed in this new form, these rules allow us to efficiently reduce or combine terms, crucial for the simplification process. Understanding these exponent rules is vital in handling powers and makes simplifying polynomials intuitive.

Rational Expressions

Rational expressions are fractions where the numerator and/or the denominator are polynomials. Simplifying these expressions often involves algebraic manipulation to remove common factors in the numerator and the denominator. In our exercise, we dealt with the rational expression:
\(\frac{(a - 2b)^2}{a+b}\).
Although it didn't allow simplification due to a lack of common factors, it was essential in the subsequent steps.
When combined with \((a-2b)^4\), it became critical to turn the division operation into a multiplication of a reciprocal - this transformed our given problem into a more simplified expression using the common factor \((a-2b)^2\).
This approach highlights how rational expressions, though initially seeming complex, can be manipulated and simplified by recognizing patterns of the components involved.
Mastering rational expressions is crucial for solving many algebraic equations and is valuable when interpreting real-world problems.