Question

Simplify the complex fraction. \(\frac{\frac{16}{x-2}}{\frac{4}{x+1}+\frac{6}{x}}\)

Short answer

The simplified form of the complex fraction is \(\frac{8x(x+1)}{5x+3(x-2)}\).

Step-by-step solution

Simplify the Denominator

To simplify the denominator, find a common denominator for both terms in the denominator, which is \(x(x+1)\). Now rewrite each term with this common denominator: \(\frac{4(x)}{x(x+1)}+\frac{6(x+1)}{x(x+1)}\). Simplify to get \( \frac{4x+6x+6}{x(x+1)} =\frac{10x+6}{x(x+1)}\)

Simplify the Entire Fraction

Now that the denominator has been simplified, rewrite the entire fraction. This would be \(\frac{\frac{16}{x-2}}{\frac{10x+6}{x(x+1)}}\). By multiplying the numerator and the denominator of the complex fraction by the reciprocal of the bottom fraction, you will remove the fraction from the denominator: \(\frac{16}{x-2} \times \frac{x(x+1)}{10x+6}\)

Simplify Resulting Fraction

When you multiply the fractions, you get \( \frac{16x(x+1)}{(x-2)(10x+6)}\). You can simplify further by dividing both the numerator and the denominator by common factors. In this case, the entire numerator can be divided by 2, and the entire denominator can also be divided by 2. This gives \(\frac{8x(x+1)}{5x+3(x-2)}\) .

Key concepts

Simplifying Fractions

Simplifying fractions is a fundamental skill in algebra that makes calculations easier and expressions more manageable. It involves reducing a fraction to its simplest form by eliminating any common factors shared by the numerator and denominator. To do this:

• Identify any common factors in both the numerator and denominator.
• Divide the numerator and denominator by their greatest common divisor (GCD).

Some crucial points to remember include:

• When simplifying, ensure that each step maintains equivalent value - the fraction remains the same in value, just looks simpler.
• Always check if further simplification is possible by re-evaluating any new form of the fraction generated.

For example, in this exercise, after simplifying, we ended at a fraction: \(\frac{8x(x+1)}{5x+3(x-2)}\). By doing so, we ensure the expression is in its simplest and most readable form.

Common Denominator

Finding a common denominator is vital when simplifying complex fractions, especially when dealing with sums or differences of fractions. A common denominator enables the addition or subtraction of different fractions by making sure their denominators are identical.To find a common denominator:

• Determine the least common multiple (LCM) of the denominators involved.
• Rewrite each fraction so its denominator is the LCM, adjusting the numerator accordingly.

In our example, the task was to simplify a denominator that had the fractions \(\frac{4}{x+1}\) and \(\frac{6}{x}\). We found the LCM, \(x(x+1)\), and rewrote each fraction with this new denominator:

• \(\frac{4(x)}{x(x+1)}\)
• \(\frac{6(x+1)}{x(x+1)}\)

Thus, we were able to effectively combine these into a single fraction \(\frac{10x+6}{x(x+1)}\), simplifying our expression's denominator.

Algebraic Fractions

Algebraic fractions contain variables within their numerators or denominators. They follow the same rules as numerical fractions but require an understanding of algebraic operations such as addition, subtraction, multiplication, and division.Key steps to remember when working with algebraic fractions include:

• Always consider the domain of the variable(s) to avoid division by zero.
• Apply factorization techniques if necessary to simplify complex expressions.

In our exercise, we dealt with the compound fraction \(\frac{\frac{16}{x-2}}{\frac{10x+6}{x(x+1)}}\). We simplified it by multiplying by the reciprocal of the denominator, transforming it into a multiplication problem that can be simplified further: \(\frac{16}{x-2} \times \frac{x(x+1)}{10x+6}\). This step is critical in reducing complexity and achieving clarity.

Factorization

Factorization is the process of breaking down algebraic expressions into simpler "factors" that, when multiplied together, give you the original expression. This is especially useful in simplifying algebraic fractions and solving equations.To factorize a given expression:

• Look for common factors in all terms.
• Apply identities like the difference of squares \((a^2 - b^2 = (a+b)(a-b))\) if applicable.
• For quadratic expressions, use methods such as splitting the middle term or the quadratic formula if needed.

In this problem, factorization played a role as we simplified fractions at each step, especially when considering common factors that allowed further simplification. For example, we reduced \(\frac{16x(x+1)}{(x-2)(10x+6)}\) to \(\frac{8x(x+1)}{5x+3(x-2)}\) by removing a factor of 2. This technique helps streamline and simply daunting algebraic problems.