Problem 45 - Algebra 1

Question

Simplify the expression. $$\frac{x-2}{x+6} \div \frac{x+8}{4 x-24} \cdot \frac{x-8}{x-2}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is 4

Step 1: Change division to multiplication

Rewrite the expression by replacing the division operation with multiplication by the reciprocal of the divisor. Hence, \(\frac{x-2}{x+6} \div \frac{x+8}{4x-24}\) becomes \(\frac{x-2}{x+6} \cdot \frac{4x-24}{x+8}\)

Step 2: Simplify the new multiplication

Now we have a multiplication of three fractions, this can be simplified to \(\frac{x-2}{x+6} \cdot \frac{4x-24}{x+8} \cdot \frac{x-8}{x-2}\). Now, notice that \(x-2\) numerator and denominator will cancel each other. This leaves us with: \(\frac{4x-24}{x+6} \cdot \frac{x-8}{x+8}\)

Step 3: Further Simplification

Also, the terms \(x+8\) in the numerator and denominator will cancel each other out. The final simplified expression is: \(\frac{4x-24}{x+6}\)

Step 4: Final Simplification

Lastly, the \(4(x-6)\) in the numerator can be divided by \(x+6\) in the denominator to give a final answer of 4, which is the much simplified form of the original expression.

Key Concepts

Rational Expressions

Rational expressions are fractions that involve polynomials in the numerator, the denominator, or both. Simplifying these expressions is similar to simplifying numerical fractions, except that one must consider the variables involved. It requires factoring polynomials and canceling common factors.

Let's take an example where we have the rational expression \( \frac{x-2}{x+6} \). In this case, both the numerator and the denominator are polynomials. Simplifying such an expression might involve factoring these polynomials to find common factors that can be canceled out. However, it's important to remember that we cannot cancel out terms, only factors; this is a common mistake students often make.

Multiplication of Fractions

When multiplying fractions, it's straightforward: simply multiply the numerators together to find the new numerator, and multiply the denominators together to find the new denominator. Unlike adding or subtracting fractions, there's no need to have a common denominator when multiplying them.

For example, multiplying \( \frac{a}{b} \) by \( \frac{c}{d} \) would result in \( \frac{a \cdot c}{b \cdot d} \). Remembering this rule can greatly simplify the process of working with rational expressions.

Algebraic Fraction Simplification

To simplify an algebraic fraction, you should first factor the polynomials if possible, then identify and cancel out any common factors. Consider any restrictions on variables that would make the denominator zero, as these values are excluded from the domain of the expression.

During our exercise, once the reciprocal was taken to switch from division to multiplication, factors such as \(x-2\) canceled out, as they appeared in both the numerator and the denominator. This process is critical in algebraic simplification because it reduces the complexity of the expression, making it easier to work with or solve.

Reciprocal of a Fraction

The reciprocal of a fraction is found by swapping the numerator and the denominator. This concept is vital when converting a division problem into a multiplication problem, as seen in our exercise when we had to divide by a fraction. For instance, the reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \) provided that \(a \eq 0\).

An important application of reciprocals is in the process of dividing fractions. To divide by a fraction, you multiply by its reciprocal. This technique was applied in step 1 of our original problem; by taking the reciprocal of the second fraction, we changed the operation from division to multiplication, which is a more straightforward process.