Problem 18 - Algebra A Combined Function

Question

Rationalize each denominator. Assume that all variables represent positive real numbers. \(\sqrt{\frac{13 a}{2 b}}\)

Step-by-Step Solution

Answer

The rationalized expression is \( \frac{\sqrt{26ab}}{2b} \).

Step 1: Identify the Denominator

The given expression is \( \sqrt{\frac{13a}{2b}} \). The denominator inside the square root is \( 2b \). We need to rationalize this denominator.

Step 2: Find the Conjugate

Since \( 2b \) is not a complex number and consists of a single term, we multiply and divide the entire expression under the square root by \( 2b \) to eliminate \( \sqrt{2b} \) from the denominator.

Step 3: Multiply by the Conjugate

Multiply \( \sqrt{\frac{13a}{2b}} \) by \( \frac{\sqrt{2b}}{\sqrt{2b}} \) to rationalize the denominator:\[ \sqrt{\frac{13a}{2b}} \times \frac{\sqrt{2b}}{\sqrt{2b}} = \frac{\sqrt{13a} \times \sqrt{2b}}{\sqrt{2b} \times \sqrt{2b}}. \]

Step 4: Simplify the Expression

Simplify the expression using the property \( \sqrt{x} \times \sqrt{x} = x \):\[ \frac{\sqrt{13a \times 2b}}{2b} = \frac{\sqrt{26ab}}{2b}. \]

Step 5: Finalized Expression

The expression \( \frac{\sqrt{26ab}}{2b} \) is the rationalized form with no square roots in the denominator.

Key Concepts

Radical Expressions

Radical expressions may seem daunting at first, but understanding them is crucial in simplifying complex mathematical formulas. At its core, a radical expression involves roots, most commonly square roots. For example, in \( \sqrt{13a} \), "13a" is under a square root, making this a radical expression.
Radical expressions often appear in various forms; thus, mastering them can help in solving numerous algebraic problems. It's essential to note that radicals can include not just numbers but also variables, making them even more versatile in problem-solving.

• They can represent a square root, cube root, or any higher order root.
• Radicals are often used to express non-exact values, such as the square root of non-perfect squares.
• Combining radicals requires specific operations, like rationalizing, to simplify them.

This versatility necessitates careful simplification to avoid errors in computations. Rationalizing is a common technique used to eliminate radicals from denominators.

Conjugates

When working with radical expressions, conjugates often come into play. A conjugate in math, typically used with complex numbers or binomials, involves reversing the sign between two terms. However, when it comes to rationalizing denominators in radicals with multiple terms, finding the conjugate can help greatly.
In this particular solution, we didn't directly use a conjugate since our denominator, \( 2b \), was a single term. However, if you face a denominator like \( a + \sqrt{b} \), you'd use its conjugate \( a - \sqrt{b} \) to rationalize the expression.
This is why conjugates are instrumental in simplifying expressions:

• Help remove radicals from denominators.
• Allow us to simplify complex expressions to a more manageable form.
• Maintain equivalency in expressions by ensuring that any multiplication or division by the conjugate respects algebraic properties.

Understanding how and when to use conjugates will lead to better proficiency in algebraic manipulations.

Simplifying Radicals

Simplifying radicals is an important skill for handling complex expressions. It involves rewriting a radical expression in its simplest form, often by factoring out perfect squares or other roots.
In our example, we simplify \( \sqrt{13a \times 2b} \) to \( \frac{\sqrt{26ab}}{2b} \). We perform this by:

• Recognizing the properties of radicals, like \( \sqrt{x} \times \sqrt{x} = x \).
• Combining terms under a single radical to streamline calculations.
• Reducing the expression after rationalizing, ensuring it is fully simplified.

This process involves meticulous attention to detail and step-by-step reductions, ensuring every expression is broken down completely. It's vital to check your work at each stage to remain accurate and precise. Mastering this technique will make tackling more advanced algebraic problems manageable.

Real Numbers

Real numbers form the backbone of almost every arithmetic operation we undertake. They include all numbers on the number line, encompassing both rational (like fractions) and irrational numbers (like roots of non-perfect squares).
The assumption that variables represent positive real numbers is crucial. It ensures that operations involving square roots do not result in undefined or imaginary values. For example, \( \sqrt{-1} \) is not possible in the realm of real numbers.
Real numbers are fundamental because:

• They ensure the solvability of equations within defined parameters.
• They make certain operations, such as root calculations, valid.
• Utilizing them appropriately helps verify the accuracy of operations and manipulations.

Understanding real numbers and their properties allows one to grasp why certain steps, such as rationalizing, are necessary and valid. Remember, each manipulation with real numbers should maintain the integrity of being a real mathematical entity.