Problem 67 - Elementary Algebra

Question

For the following problems, perform the multiplications and divisions. $$ \frac{2 a-b}{a+b} \cdot \frac{a+3 b}{a-5 b} \cdot \frac{a-5 b}{2 a-b} $$

Step-by-Step Solution

Answer

Question: Perform the multiplications and divisions of the given expression and simplify it: $$ \frac{2a - b}{a + b} \cdot \frac{a + 3b}{a - 5b} \cdot \frac{a - 5b}{2a - b} $$ Answer: The simplified expression of the given operation is: $$ \frac{a + 3b}{a + b} $$

Step 1: Write the expression

Write down the given expression: $$ \frac{2a - b}{a + b} \cdot \frac{a + 3b}{a - 5b} \cdot \frac{a - 5b}{2a - b} $$

Step 2: Multiply the numerators together

Multiply the numerators of the three fractions together: \((2a - b)(a + 3b)(a - 5b)\)

Step 3: Multiply the denominators together

Multiply the denominators of the three fractions together: \((a + b)(a - 5b)(2a - b)\)

Step 4: Identify common factors

Identify the common factors in both numerators and denominators: Notice that \((2a - b)\) is a common factor in both the numerator and the denominator, and (\((a - 5b)\) is also a common factor in both the numerator and the denominator.

Step 5: Simplify the expression by canceling out common factors

Cancel out the common factors between the numerators and the denominators: $$ \frac{(2a - b)(a + 3b)(a - 5b)}{(a + b)(a - 5b)(2a - b)} = \frac{\cancel{(2a - b)}(a + 3b)\cancel{(a - 5b)}}{(a + b)\cancel{(a - 5b)}\cancel{(2a - b)}} $$

Step 6: Simplify

Rewrite the expression after canceling out the common factors: $$ \frac{a + 3b}{a + b} $$ So the simplified expression of the given operation is: $$ \frac{a + 3b}{a + b} $$

Key Concepts

Multiplying Fractions in Algebra

Multiplying fractions in algebra follows the same principles as multiplying numerical fractions. In general, the process involves multiplying the numerators together to find the new numerator, and multiplying the denominators together for the new denominator. For example, to multiply two algebraic fractions, such as \( \frac{a}{b} \) and \( \frac{c}{d} \), you would perform the following operation: \( \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \).

When dealing with algebraic expressions, it's important to handle the variables with care, ensuring that like terms are properly combined when multiplying the expressions. For instance:\[ (x + y)(x - y) = x^2 - y^2 \]
This step is vital in simplifying algebraic fractions, as it can lead to identifying common factors that may be canceled out in the simplification process.

Canceling Common Factors

Canceling common factors between the numerator and the denominator is a crucial step in simplifying algebraic expressions. This process is akin to reducing fractions to their simplest form in basic arithmetic. Common factors in algebraic expressions are terms that appear both in the numerator and the denominator. For example, if you have \( \frac{x^2y}{xy^2} \), you can cancel the common factor of \( xy \) to simplify the expression to \( \frac{x}{y} \).

However, it's important to watch out for variables and ensure that you are only canceling terms that are identical and not merely similar. Simplifying algebraic fractions often requires factoring expressions to reveal common factors that are not immediately visible.

• Always factor expressions fully before attempting to cancel.
• Only cancel factors, not terms that are added or subtracted.
• Check for common factors that might be complex expressions themselves.

By attentively canceling common factors, you can greatly simplify algebraic fractions and make them easier to work with.

Algebraic Fractions Simplification

The end goal of operating with algebraic fractions is often to simplify the expression. Simplification can involve several steps: expanding expressions, factoring, finding and canceling common factors, and combining like terms. To fully simplify an algebraic fraction, you may perform a combination of these methods.

For instance, in an expression like \( \frac{2 a-b}{a+b} \times \frac{a+3 b}{a-5 b} \times \frac{a-5 b}{2 a-b} \), simplification includes:

• Multiplying the fractions as detailed in the first section, focusing on the numerators and denominators separately.
• Factoring to identify and cancel common factors, as mentioned in the second section.
• Reducing the expression to its simplest form, which may mean eliminating all common factors and rewriting the expression with the remaining terms.

By carefully going through each step and checking that all common factors are canceled, the expression is vastly simplified, resulting in an easier to understand and more manageable algebraic fraction.