Problem 38 - Algebra A Combined Function

Question

Add or subtract as indicated. Assume that all variables represent positive real numbers. $$ \frac{\sqrt[4]{48}}{5 x}-\frac{2 \sqrt[4]{3}}{10 x} $$

Step-by-Step Solution

Answer

\(\frac{\sqrt[4]{3}}{5x}\) is the simplified result.

Step 1: Factor Out Common Terms

First, observe that both terms in the expression have denominators that involve **\(x\)**. Since the denominators are similar, identify any possible factorization opportunity in the numerators.

Step 2: Simplify Numerators Using Radical Properties

Let's simplify the numerators. Note that \( \sqrt[4]{48} \) can be rewritten using its prime factors:\[\sqrt[4]{48} = \sqrt[4]{16 \times 3} = \sqrt[4]{16} \times \sqrt[4]{3} = 2 \sqrt[4]{3}\]. This gives us:\[\frac{2\sqrt[4]{3}}{5x} - \frac{2\sqrt[4]{3}}{10x}\].

Step 3: Create Common Denominator

The common denominator between \(5x\) and \(10x\) is \(10x\). Convert the first fraction to have this common denominator:\[\frac{2\sqrt[4]{3}}{5x} = \frac{2\sqrt[4]{3} \cdot 2}{10x} = \frac{4\sqrt[4]{3}}{10x}\].

Step 4: Combine Fractions

Now, subtract the fractions:\[\frac{4\sqrt[4]{3}}{10x} - \frac{2\sqrt[4]{3}}{10x} = \frac{(4\sqrt[4]{3} - 2\sqrt[4]{3})}{10x} = \frac{2\sqrt[4]{3}}{10x}\].

Step 5: Simplify the Result

Factor out the common \(2\) from the numerator:\[\frac{2\sqrt[4]{3}}{10x} = \frac{\sqrt[4]{3}}{5x}\]. This is the final simplified expression.

Key Concepts

Simplification

Simplification is the process of making an algebraic expression or fraction easier to understand or solve. In the context of algebraic fractions, it often involves reducing fractions or simplifying expressions by finding and canceling out common terms.

• Identify like terms: Look for terms that are similar in structure or that can be simplified by combinatorial properties.
• Reduce fractions: If both the numerator and the denominator share a common factor, we can simplify the fraction by dividing both of them by that common factor.
• Rewrite expressions: Expression simplification can also involve expanding or factoring numerators or denominators using algebraic principles and properties, like distributing or factoring.

In the solution provided, simplification occurs when rewriting the numerators using radical properties and then further simplifying the final fraction by factoring out common terms.

Common Denominator

Finding a common denominator is crucial when adding or subtracting fractions. It allows us to express multiple fractions with the same denominator, which is necessary for their arithmetic combination.

• Identify the Least Common Multiple (LCM): The least common multiple of the denominators provides the smallest common denominator possible for fractions.
• Rewrite each fraction: Adjust each fraction so that its denominator is the common denominator. This often involves multiplying the numerator and denominator by the same factor.
• Combine the Fractions: Once the fractions have common denominators, the numerators can be added or subtracted directly.

In the exercise, the step involving finding the common denominator was key to subtracting the fractions efficiently. This process unified the denominators of both fractions to 10x, allowing subtraction to occur easily.

Radicals

Radicals are expressions that involve roots, such as square roots or fourth roots, as seen in the exercise with symbols like \(\sqrt[4]{48}\). Working with radicals often requires simplifying them, which may involve finding perfect powers or using prime factorization.

• Simplify radicals: Break down the number under the radical to its prime factors and determine if it can be simplified, as was done with \(\sqrt[4]{48} = \sqrt[4]{16 \times 3} = 2\sqrt[4]{3}\).
• Rational expressions: Operations with radicals often involve simplifying the radicand or managing radical expressions by adopting these simplification techniques.

In the problem's step-by-step solution, rewriting radicals helps simplify the fractions' numerators, making subtraction feasible.

Prime Factorization

Prime factorization breaks down a number into a product of its prime factors. This is particularly useful when simplifying radicals or finding the greatest common divisor or least common multiple.

• Identify prime factors: Factor numbers down into primes, such as expressing 48 as \(2^4 \times 3\).
• Simplify radicals: Use prime factorization to simplify the expression within the radical, helping to break down complex expressions into simpler components.
• Common Factors: In simplifying expressions, knowing the prime factors can help identify common terms to cancel or simplify during fraction reduction.

In our exercise, simplifying \(\sqrt[4]{48}\) to \(2\sqrt[4]{3}\) by using its prime factors demonstrates an effective use of prime factorization.