Problem 24 - Algebra A Combined Function

Question

Rationalize each denominator. Assume that all variables represent positive real numbers. \(\frac{1}{\sqrt{32 x}}\)

Step-by-Step Solution

Answer

\( \frac{\sqrt{32x}}{32x} \)

Step 1: Find the Conjugate

To rationalize the denominator, we first need to identify a form that eliminates the square root. Since there is only a single square root term, the conjugate is simply the square root itself, \( \sqrt{32x} \). We multiply both the numerator and the denominator by this term.

Step 2: Multiply by the Conjugate

Multiply the fraction \( \frac{1}{\sqrt{32x}} \) by \( \frac{\sqrt{32x}}{\sqrt{32x}} \) to eliminate the square root from the denominator:\[\frac{1 \cdot \sqrt{32x}}{\sqrt{32x} \cdot \sqrt{32x}} = \frac{\sqrt{32x}}{32x}\]

Step 3: Simplify the Expression

Now, simplify the expression:The numerator remains \( \sqrt{32x} \). The denominator is \( \sqrt{32x} \cdot \sqrt{32x} = 32x \) as squaring the square root cancels it out.Thus, the expression is \( \frac{\sqrt{32x}}{32x} \).

Key Concepts

Square Roots

A square root is a mathematical operation that involves finding a number, which when multiplied by itself, yields the original number. For example, the square root of 9 is 3, because 3 multiplied by 3 gives 9. In short, \[\sqrt{9} = 3.\]Understanding square roots is crucial, especially when dealing with expressions and equations in algebra. In the context of our problem, the square root appears in the denominator as \(\sqrt{32x}\).

• To "rationalize" the denominator means eliminating the square root from the bottom of the fraction.
• This helps simplify mathematical expressions because dealing with whole numbers or other simpler forms makes calculations easier and more manageable.

Rationalizing involves multiplying the expression by a form of 1 that doesn't change its value. In other words, we use \(\frac{\sqrt{32x}}{\sqrt{32x}}\), which keeps the original value intact while accomplishing the task of removing the square root from the denominator.

Simplifying Expressions

Simplifying an expression involves reducing it to its most concise and clear form without changing its value. In our problem, \(\frac{1}{\sqrt{32x}}\), we aim to make the denominator a rational number, resulting in a more manageable expression. By multiplying both the numerator and denominator by \(\sqrt{32x}\), we "clear" the square root:

• This process transforms \(\frac{1}{\sqrt{32x}}\) into \(\frac{\sqrt{32x}}{32x}\).
• In the numerator, multiplication introduces the square root, making it \(\sqrt{32x}\).
• The denominator simplifies to \(32x\) because \(\sqrt{32x} \times \sqrt{32x} = 32x\).

The simplified expression \(\frac{\sqrt{32x}}{32x}\) is now more straightforward and free of irrational numbers in the denominator.

Positive Real Numbers

In mathematics, real numbers include all the numbers on the number line, encompassing both rational and irrational numbers. Positive real numbers are those greater than zero. Understanding that variables in an expression represent positive real numbers is essential, particularly when dealing with operations like square roots.

• Positive real numbers ensure the square root operations can be performed without resulting in imaginary numbers.
• This assumption simplifies the problem, as the variables won't lead to undefined or complex results in expressions.
• If a variable represents a positive real number, you don't face the issues associated with taking square roots of negative numbers.

By assuming that all variables represent positive real numbers, we ensure the rationalizing process is straightforward and leads to valid, coherent expressions. This understanding is vital when simplifying mathematical expressions, making calculations accessible and results predictable.