Problem 49 - Algebra 2

Question

Simplify each expression. $$ \frac{8^{\frac{1}{6}}-9^{\frac{1}{4}}}{\sqrt{3}+\sqrt{2}} $$

Step-by-Step Solution

Answer

\(8^{\frac{1}{6}}\sqrt{3} - 8^{\frac{1}{6}}\sqrt{2} - 9^{\frac{1}{4}}\sqrt{3} + 9^{\frac{1}{4}}\sqrt{2}\)

Step 1: Identify Components

First, identify the separate components of the expression. The numerator is \(8^{\frac{1}{6}} - 9^{\frac{1}{4}}\) and the denominator is \(\sqrt{3} + \sqrt{2}\).

Step 2: Simplify Each Term

Simplify each component if possible. We can't extract simpler forms from \(8^{\frac{1}{6}}\) and \(9^{\frac{1}{4}}\) directly without additional transformations.

Step 3: Rationalize the Denominator

Multiply both the numerator and denominator by the conjugate of the denominator to rationalize it. The conjugate of \(\sqrt{3} + \sqrt{2}\) is \(\sqrt{3} - \sqrt{2}\).This gives:\[\frac{(8^{\frac{1}{6}}-9^{\frac{1}{4}})(\sqrt{3} - \sqrt{2})}{(\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2})}\]

Step 4: Simplify the Denominator

Simplify the product of the denominators:\[(\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2}) = 3 - 2 = 1\]

Step 5: Expand the Numerator

Expand \((8^{\frac{1}{6}}-9^{\frac{1}{4}})(\sqrt{3} - \sqrt{2})\):\[8^{\frac{1}{6}}\sqrt{3} - 8^{\frac{1}{6}}\sqrt{2} - 9^{\frac{1}{4}}\sqrt{3} + 9^{\frac{1}{4}}\sqrt{2}\]

Step 6: Solution in Simplified Form

Since the denominator simplifies to 1, the expression simplifies to the numerator:\[8^{\frac{1}{6}}\sqrt{3} - 8^{\frac{1}{6}}\sqrt{2} - 9^{\frac{1}{4}}\sqrt{3} + 9^{\frac{1}{4}}\sqrt{2}\]

Key Concepts

Simplifying Radical Expressions

Simplifying radical expressions involves dealing with the roots and simplifying them to their simplest form. It is a critical skill that helps make complex expressions more manageable. In the original exercise, we encounter radicals such as \(\sqrt{3}\) and \(\sqrt{2}\). These are square roots, which are considered radicals.

When simplifying, check if the radical expression can be broken down further. Factors of perfect squares should be identified, allowing for further reduction. For instance, \(\sqrt{12}\) can be simplified to \(2\sqrt{3}\), because \(12 = 4 \times 3\), and \(\sqrt{4} = 2\).

While the radicals \(\sqrt{3}\) and \(\sqrt{2}\) in our expression cannot be simplified further, understanding this process is crucial for tackling similar problems involving radicals. It's essential to practice recognizing when a radical expression is fully simplified to avoid overcomplicating calculations.

Exponents

Exponents indicate how many times a number is multiplied by itself. They can transform significantly the way we handle numbers, especially if they are fractions. In the original expression, we see fractional exponents: \(8^{\frac{1}{6}}\) and \(9^{\frac{1}{4}}\).

Fractional exponents represent roots. For example, \(a^{\frac{1}{n}}\) is equivalent to the \(n\)-th root of \(a\). Thus, \(8^{\frac{1}{6}}\) is the sixth root of 8, and \(9^{\frac{1}{4}}\) is the fourth root of 9. This helps simplify problems, especially when comparing or combining expressions with different bases.

Understanding the properties of exponents can be extremely helpful:

• \(a^{m}\times a^{n} = a^{m+n}\)
• \((a^{m})^{n} = a^{m\times n}\)
• \(a^{-n} = \frac{1}{a^n}\)

With these rules, expressions can be rewritten and simplified, assisting in complex calculations seen in many algebraic problems.

Conjugate Pairs

The concept of conjugate pairs is essential in rationalizing the denominator of an expression involving radicals. A conjugate consists of the same two terms but with the opposite operation between them. In the exercise, the conjugate of \(\sqrt{3} + \sqrt{2}\) is \(\sqrt{3} - \sqrt{2}\).

Multiplying a numerator and a denominator by a conjugate can help eliminate the radical in the denominator. This process is called rationalizing the denominator. The product of a binomial and its conjugate results in a difference of squares. Using the formula \((a + b)(a - b) = a^2 - b^2\), you effectively remove the radical from the denominator when it becomes a perfect square.

In the given problem, applying the conjugate simplifies

• The denominator \((\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2})\) which results in \(3 - 2 = 1\), thereby eliminating the radical.
• With a denominator of 1, the expression simplifies drastically to simply the expanded numerator.

This technique is invaluable when simplifying complex fractions involving radicals and enables more effortless manipulation and calculation of expressions.